Vector & 3D Formula Compendium

JEE ADV

Made by -UnknownShadow08

⟶ Vector Algebra 11 SECTIONS
◈ Basics, Magnitude & Unit Vectors 6 ▶
Magnitude of Vector
\[\left|\vec{A}\right| = \sqrt{a_1^2 + a_2^2 + a_3^2}\]
Find the length/modulus of any vector given in component form.
Unit Vector
\[\hat{a} = \frac{\vec{a}}{|\vec{a}|}\]
Get direction of a vector with magnitude 1; used in projections, perpendicular vectors.
Vector in Space
\[\vec{r} = x\hat{i} + y\hat{j} + z\hat{k},\quad |\vec{r}| = \sqrt{x^2+y^2+z^2}\]
Standard component form of a 3D vector from origin to point P(x,y,z).
Position Vector of AB
\[\overrightarrow{AB} = \vec{b} - \vec{a} = \text{PV of } B - \text{PV of } A\]
Express a vector between two points using their position vectors.
Vector of Given Magnitude in Direction d̂
\[\vec{B} = |\vec{B}| \times \hat{d} = |\vec{B}| \cdot \frac{\vec{d}}{|\vec{d}|}\]
Construct a vector of given magnitude in a specified direction.
Scalar Multiplication Properties
\[m(\vec{a}) = \vec{a}m = m\vec{a}\]
\[m(n\vec{a}) = n(m\vec{a}) = (mn)\vec{a}\]
\[(m+n)\vec{a} = m\vec{a} + n\vec{a},\quad m(\vec{a}+\vec{b}) = m\vec{a}+m\vec{b}\]
Distribute/factor scalars in vector expressions.
◈ Addition, Subtraction & Laws 7 ▶
Triangle Law of Addition
\[\overrightarrow{OB} = \overrightarrow{OA} + \overrightarrow{AB} = \vec{a} + \vec{b}\]
Add two vectors placed head-to-tail; resultant is closing side.
Parallelogram Law
\[\overrightarrow{OP} + \overrightarrow{OQ} = \overrightarrow{OR} \implies \vec{a}+\vec{b} = \overrightarrow{OR}\]
Add two co-initial vectors; resultant is the diagonal through that point.
Polygon Law
\[\vec{a}+\vec{b}+\vec{c}+\vec{d}+\vec{e} = \overrightarrow{OE}\]
\[\text{If polygon is closed} \Rightarrow \sum\vec{a} = \vec{0}\]
Sum of vectors forming a closed polygon is zero (e.g. regular hexagon opposite vectors).
Properties of Vector Addition
\[\vec{a}+\vec{b} = \vec{b}+\vec{a} \quad\text{(commutative)}\]
\[(\vec{a}+\vec{b})+\vec{c} = \vec{a}+(\vec{b}+\vec{c}) \quad\text{(associative)}\]
\[\vec{a}+\vec{0} = \vec{a} \quad\text{(additive identity)}\]
\[\vec{a}+(-\vec{a}) = \vec{0} \quad\text{(additive inverse)}\]
Rearranging sums of multiple vectors; proving vector identities.
Vector Subtraction
\[\vec{a} - \vec{b} = \vec{a} + (-\vec{b})\]
Subtraction is addition of negative; does NOT satisfy commutative or associative law.
Geometric Interpretation: |a+b| = |a−b|
\[|\vec{a}+\vec{b}| = |\vec{a}-\vec{b}| \iff \vec{a} \perp \vec{b}\]
Diagonals of a parallelogram are equal iff it is a rectangle; test perpendicularity.
Midpoint of Median via Triangle Law
\[\overrightarrow{DE} = \frac{1}{2}\overrightarrow{BC} \implies DE \parallel BC,\ DE = \frac{BC}{2}\]
Prove midpoint theorem: line joining midpoints of two sides is half and parallel to third.
◈ Section Formula, Centroid & Collinearity 9 ▶
Internal Division
\[\overrightarrow{OC} = \vec{r} = \frac{m\vec{b} + n\vec{a}}{m+n}\]
Find position vector of point dividing AB in ratio m:n internally.
External Division
\[\overrightarrow{OC} = \vec{r} = \frac{m\vec{b} - n\vec{a}}{m-n}\]
Find position vector of point dividing AB in ratio m:n externally.
Midpoint Formula
\[\text{Midpoint of } AB = \frac{\vec{a}+\vec{b}}{2}\]
Used constantly in geometry proofs (medians, diagonals bisecting).
Centroid of Triangle
\[G = \frac{\vec{a}+\vec{b}+\vec{c}}{3}\]
Point where three medians meet; divides each median in 2:1 from vertex.
Centroid of Tetrahedron
\[\vec{g} = \frac{\vec{a}+\vec{b}+\vec{c}+\vec{d}}{4}\]
Centre where vertex-to-opposite-face-centroid lines concur; also centre of parallelepiped.
Collinear Vectors (Symbolic Condition)
\[\vec{a} \text{ and } \vec{b} \text{ collinear} \iff \vec{a} = K\vec{b},\ K \in \mathbb{R}\]
Test or prove two vectors are parallel/collinear using a single scalar.
Collinear Points (3 Points)
\[A,B,C \text{ collinear} \iff \overrightarrow{AB} = \lambda\overrightarrow{BC}\]
\[\iff \exists\ x,y,z \text{ (not all zero): } x\vec{a}+y\vec{b}+z\vec{c}=\vec{0},\ x+y+z=0\]
Necessary & sufficient condition for three points to be collinear.
Euler Line (Circumcenter S, Orthocenter O)
\[\vec{G} = \frac{\vec{O}+2\vec{S}}{3}\]
\[\overrightarrow{SA}+\overrightarrow{SB}+\overrightarrow{SC} = \overrightarrow{SO}\]
Relate circumcenter S, centroid G, and orthocenter O of a triangle.
4 Points Coplanar (N&S Condition)
\[x\vec{a}+y\vec{b}+z\vec{c}+t\vec{d}=\vec{0},\quad x+y+z+t=0\]
Check if four position vectors lie in one plane.
◈ Vector Equation of Lines & Bisectors 5 ▶
Line Through Point Parallel to Vector
\[\vec{r} = \vec{a} + t\vec{b}, \quad t \in \mathbb{R}\]
Write equation of a line when one point and direction are known.
Line Through Two Points
\[\vec{r} = \vec{a} + t(\vec{b}-\vec{a}) = (1-t)\vec{a} + t\vec{b}\]
Parametric equation of line joining points with PV's a⃗ and b⃗.
Angle Bisector Direction
\[\text{Bisector direction}: \hat{a}+\hat{b} \quad (\lambda>0)\]
Direction of the bisector between two vectors (from their unit vectors).
Two Angle Bisectors of Two Lines
\[\vec{r} = \vec{a} + t(\hat{b}+\hat{c}) \quad\text{(one bisector)}\]
\[\vec{r} = \vec{a} + s(\hat{c}-\hat{b}) \quad\text{(other bisector)}\]
Find equations of both angle bisector lines at a given point between two lines.
Skew / Parallel / Intersecting Lines
\[\text{SD}=0 \Rightarrow \text{intersecting (coplanar)}\]
\[\text{Two lines intersect OR parallel} \Rightarrow \text{coplanar}\]
\[\text{Parallel with common point} \Rightarrow \text{coincident}\]
Classify spatial lines before computing intersection or shortest distance.
◈ Dot Product (Scalar Product) 12 ▶
Definition
\[\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta, \quad 0 \le \theta \le \pi\]
Fundamental formula; gives a scalar; positive for acute, negative for obtuse angle.
Self Dot Product
\[\vec{a}\cdot\vec{a} = |\vec{a}|^2 = a^2\]
Find squared magnitude of a vector from its dot product with itself.
Properties
\[\vec{a}\cdot\vec{b} = \vec{b}\cdot\vec{a} \quad\text{(commutative)}\]
\[\vec{a}\cdot(\vec{b}+\vec{c}) = \vec{a}\cdot\vec{b}+\vec{a}\cdot\vec{c} \quad\text{(distributive)}\]
Simplify complex dot product expressions by distributing.
Perpendicularity Condition
\[\vec{a}\cdot\vec{b} = 0 \iff \vec{a}\perp\vec{b} \quad (\vec{a},\vec{b}\ne\vec{0})\]
Prove or test orthogonality of two vectors (altitudes, normals, etc.).
Standard Basis Results
\[\hat{i}\cdot\hat{i}=\hat{j}\cdot\hat{j}=\hat{k}\cdot\hat{k}=1\]
\[\hat{i}\cdot\hat{j}=\hat{j}\cdot\hat{k}=\hat{k}\cdot\hat{i}=0\]
Evaluate component-wise dot products; foundation for the general formula.
General Formula & Angle
\[\vec{a}\cdot\vec{b} = a_1b_1+a_2b_2+a_3b_3\]
\[\cos\theta = \frac{a_1b_1+a_2b_2+a_3b_3}{\sqrt{a_1^2+a_2^2+a_3^2}\cdot\sqrt{b_1^2+b_2^2+b_3^2}}\]
Compute angle between vectors given components; workhorse formula for JEE.
Key Algebraic Identities (6)
\[(\vec{a}+\vec{b})\cdot(\vec{a}-\vec{b}) = a^2-b^2\]
\[(\vec{a}+\vec{b})^2 = a^2+2\vec{a}\cdot\vec{b}+b^2\]
\[(\vec{a}-\vec{b})^2 = a^2-2\vec{a}\cdot\vec{b}+b^2\]
\[(\vec{a}+\vec{b})^2 = (\vec{a}-\vec{b})^2 + 4\vec{a}\cdot\vec{b}\]
\[(\vec{a}+\vec{b}+\vec{c})^2 = a^2+b^2+c^2+2\textstyle\sum\vec{a}\cdot\vec{b}\]
\[\vec{a}\cdot\vec{b} = \tfrac{1}{4}\!\left[(\vec{a}+\vec{b})^2-(\vec{a}-\vec{b})^2\right]\]
Expand/simplify magnitude expressions; get dot product from sum/difference magnitudes.
Projection of a⃗ on b⃗
\[\text{Scalar projection} = \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|} = \vec{a}\cdot\hat{b}\]
\[\text{Vector component along }\vec{b} = \left(\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|^2}\right)\vec{b} = (\vec{a}\cdot\hat{b})\hat{b}\]
\[\text{Component}\perp\vec{b} = \vec{a} - \left(\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|^2}\right)\vec{b}\]
Resolve a vector along and perpendicular to a given direction; foot of perpendicular.
Minimum Value / Cauchy-Schwarz Application
\[(3\hat{i}+4\hat{j}+12\hat{k})\cdot(a\hat{i}+b\hat{j}+c\hat{k}) \le 13\sqrt{a^2+b^2+c^2}\]
\[\Rightarrow \max(pa+qb+rc) = \sqrt{p^2+q^2+r^2}\cdot\sqrt{a^2+b^2+c^2}\]
Maximise linear expressions subject to constraint on magnitude (optimisation problems).
Fundamental Theorem in Plane
\[\vec{a},\vec{b}\ \text{non-collinear} \Rightarrow \vec{r} = x\vec{a}+y\vec{b}\ \text{(unique)}\]
Any coplanar vector can be uniquely expressed in terms of two non-collinear base vectors.
Fundamental Theorem in Space
\[\vec{a},\vec{b},\vec{c}\ \text{non-coplanar} \Rightarrow \vec{r}=x\vec{a}+y\vec{b}+z\vec{c}\ \text{(unique)}\]
Basis of 3D space; every vector is a unique linear combination of three non-coplanar vectors.
Minimum of |â+b̂+ĉ|²
\[\hat{a}\cdot\hat{b}+\hat{b}\cdot\hat{c}+\hat{c}\cdot\hat{a} \ge -\tfrac{3}{2}\]
\[|\hat{a}+\hat{b}+\hat{c}|^2 \ge 0 \Rightarrow \min|\hat{a}+\hat{b}+\hat{c}|^2 = 3\]
Find least value of sum of squared magnitudes for unit vectors.
◈ Cross Product (Vector Product) 11 ▶
Definition
\[\vec{a}\times\vec{b} = |\vec{a}||\vec{b}|\sin\theta\,\hat{n}\]
n̂ is perpendicular to both via right-hand screw rule; result is a vector.
Unit Vector ⊥ to Plane of a⃗, b⃗
\[\hat{n} = \pm\frac{\vec{a}\times\vec{b}}{|\vec{a}\times\vec{b}|}\]
Find normal to a plane defined by two vectors; used in plane equations.
Vector of Magnitude r ⊥ to Plane
\[\vec{v} = \pm\frac{r(\vec{a}\times\vec{b})}{|\vec{a}\times\vec{b}|}\]
Construct a vector of specific magnitude perpendicular to a given plane.
Sine of Angle Between Vectors
\[\sin\theta = \frac{|\vec{a}\times\vec{b}|}{|\vec{a}||\vec{b}|}\]
Find angle between vectors when cross product is known; complement to cosθ via dot product.
Lagrange's Identity
\[|\vec{a}\times\vec{b}|^2 = a^2b^2-(\vec{a}\cdot\vec{b})^2 = \begin{vmatrix}\vec{a}\cdot\vec{a} & \vec{a}\cdot\vec{b}\\\vec{b}\cdot\vec{a} & \vec{b}\cdot\vec{b}\end{vmatrix}\]
Compute |a×b| from dot products alone; avoids computing the cross product explicitly.
Properties
\[\vec{a}\times\vec{b}=\vec{0} \iff \vec{a}=\lambda\vec{b}\ \text{(parallel/collinear)}\]
\[\vec{a}\times\vec{b} \ne \vec{b}\times\vec{a}\ \text{(not commutative)}\]
\[\vec{a}\times(\vec{b}+\vec{c}) = \vec{a}\times\vec{b}+\vec{a}\times\vec{c}\ \text{(distributive)}\]
\[(\vec{a}\times\vec{b})\cdot\vec{a}=0,\quad (\vec{a}\times\vec{b})\cdot\vec{b}=0\]
Simplify cross product expressions; confirm perpendicularity of result.
Standard Basis Cross Products
\[\hat{i}\times\hat{i}=\hat{j}\times\hat{j}=\hat{k}\times\hat{k}=\vec{0}\]
\[\hat{i}\times\hat{j}=\hat{k},\quad \hat{j}\times\hat{k}=\hat{i},\quad \hat{k}\times\hat{i}=\hat{j}\]
Evaluate cross products of unit vectors; cyclic order i→j→k→i gives +, reverse gives −.
Determinant Formula
\[\vec{a}\times\vec{b} = \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}\]
Compute cross product in component form; standard method for JEE calculations.
Area of Parallelogram
\[\text{Area} = |\vec{a}\times\vec{b}|\]
\[\text{Using diagonals } \vec{d}_1, \vec{d}_2: \text{Area} = \tfrac{1}{2}|\vec{d}_1\times\vec{d}_2|\]
Area of parallelogram with adjacent sides a⃗, b⃗ or known diagonals.
Area of Triangle
\[\text{Area} = \frac{1}{2}|\vec{a}\times\vec{b}| = \frac{1}{2}|\overrightarrow{BA}\times\overrightarrow{BC}| = \frac{1}{2}|\overrightarrow{AB}\times\overrightarrow{AC}|\]
Direct formula for area given two sides as vectors.
Vector Area of △ABC (Position Vectors)
\[\vec{\Delta} = \frac{1}{2}\left[(\vec{c}-\vec{b})\times(\vec{a}-\vec{b})\right] = \frac{1}{2}[(\vec{a}\times\vec{b})+(\vec{b}\times\vec{c})+(\vec{c}\times\vec{a})]\]
\[\text{If collinear: } \vec{a}\times\vec{b}+\vec{b}\times\vec{c}+\vec{c}\times\vec{a}=\vec{0}\]
Area using position vectors of vertices; collinearity test using cross products.
◈ Shortest Distance Between Lines 3 ▶
SD Between Two Skew Lines
\[d = \frac{|(\vec{a}_2-\vec{a}_1)\cdot(\vec{b}_1\times\vec{b}_2)|}{|\vec{b}_1\times\vec{b}_2|}\]
Minimum distance between non-parallel, non-intersecting (skew) lines in 3D.
SD Between Two Parallel Lines
\[d = \frac{|\vec{b}\times(\vec{a}_2-\vec{a}_1)|}{|\vec{b}|}\]
Distance between two parallel lines with common direction b⃗, through points a⃗₁ and a⃗₂.
SD = 0 Interpretation
\[d = 0 \Rightarrow \text{Lines intersect (coplanar)}\]
\[\text{Coplanarity test: } (\vec{a}_2-\vec{a}_1)\cdot(\vec{b}_1\times\vec{b}_2) = 0\]
Check if two lines intersect before finding their point of intersection.
◈ Scalar Triple Product (Box Product) 10 ▶
Definition
\[[\vec{a}\ \vec{b}\ \vec{c}] = \vec{a}\cdot(\vec{b}\times\vec{c}) = (\vec{a}\times\vec{b})\cdot\vec{c} = |\vec{a}||\vec{b}||\vec{c}|\sin\theta\cos\phi\]
Scalar result of a dot product followed by a cross product; gives signed volume.
Determinant Formula
\[[\vec{a}\ \vec{b}\ \vec{c}] = \begin{vmatrix}a_1&a_2&a_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{vmatrix}\]
Compute box product from components; standard computational method.
Cyclic Property
\[[\vec{a}\ \vec{b}\ \vec{c}] = [\vec{b}\ \vec{c}\ \vec{a}] = [\vec{c}\ \vec{a}\ \vec{b}]\]
\[[\vec{a}\ \vec{b}\ \vec{c}] = -[\vec{a}\ \vec{c}\ \vec{b}]\quad\text{(swap reverses sign)}\]
Rearrange vectors in box product; swapping any two changes sign.
Special Values
\[[\hat{i}\ \hat{j}\ \hat{k}] = 1\]
\[[K\vec{a}\ \vec{b}\ \vec{c}] = K[\vec{a}\ \vec{b}\ \vec{c}]\]
\[[(\vec{a}+\vec{b})\ \vec{c}\ \vec{d}] = [\vec{a}\ \vec{c}\ \vec{d}] + [\vec{b}\ \vec{c}\ \vec{d}]\]
Scale and linearity properties; simplify box products with sums of vectors.
Volume of Parallelepiped
\[V = |[\vec{a}\ \vec{b}\ \vec{c}]|\]
Volume of the parallelepiped formed by three coterminous edge vectors.
Volume of Tetrahedron
\[V = \frac{1}{6}|[\vec{a}\ \vec{b}\ \vec{c}]|\]
Volume of tetrahedron OABC where a⃗,b⃗,c⃗ are PV's of A,B,C from O.
Coplanarity Condition
\[\vec{a},\vec{b},\vec{c}\ \text{coplanar} \iff [\vec{a}\ \vec{b}\ \vec{c}] = 0\]
Test if three vectors are linearly dependent / lie in the same plane.
Important Identity 1
\[[\vec{a}-\vec{b}\ \ \vec{b}-\vec{c}\ \ \vec{c}-\vec{a}] = 0\]
Vectors formed by cyclic differences of three vectors are always coplanar.
Important Identity 2
\[[\vec{a}+\vec{b}\ \ \vec{b}+\vec{c}\ \ \vec{c}+\vec{a}] = 2[\vec{a}\ \vec{b}\ \vec{c}]\]
Box product of pairwise sums equals twice the original; used in geometry proofs.
Reciprocal System Result
\[[\vec{a}\times\vec{b}\ \ \vec{b}\times\vec{c}\ \ \vec{c}\times\vec{a}] = [\vec{a}\ \vec{b}\ \vec{c}]^2\]
Box product of cross products equals square of original box product; used in reciprocal bases.
◈ Vector Triple Product 3 ▶
BAC–CAB Rule
\[\vec{a}\times(\vec{b}\times\vec{c}) = (\vec{a}\cdot\vec{c})\vec{b} - (\vec{a}\cdot\vec{b})\vec{c}\]
Expand a nested cross product into a linear combination; key manipulation tool.
Reversed Order
\[(\vec{a}\times\vec{b})\times\vec{c} = (\vec{a}\cdot\vec{c})\vec{b} - (\vec{b}\cdot\vec{c})\vec{a}\]
Note: a×(b×c) ≠ (a×b)×c in general; parentheses matter.
Geometric Interpretation
\[\vec{a}\times(\vec{b}\times\vec{c}) \text{ lies in the plane of } \vec{b},\vec{c}\text{ and }\perp\vec{a}\]
\[(\vec{a}\times\vec{b})\times\vec{c} \text{ lies in the plane of } \vec{a},\vec{b}\text{ and }\perp\vec{c}\]
Understand the geometric direction of the result; helps in checking answers.
◈ Products of Four Vectors 3 ▶
Scalar Product of Four Vectors
\[(\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d}) = (\vec{a}\cdot\vec{c})(\vec{b}\cdot\vec{d}) - (\vec{a}\cdot\vec{d})(\vec{b}\cdot\vec{c}) = \begin{vmatrix}\vec{a}\cdot\vec{c}&\vec{a}\cdot\vec{d}\\\vec{b}\cdot\vec{c}&\vec{b}\cdot\vec{d}\end{vmatrix}\]
Convert product of cross products to dot products; avoids explicit cross product computation.
Vector Product of Four Vectors
\[(\vec{a}\times\vec{b})\times(\vec{c}\times\vec{d}) = [\vec{a}\vec{b}\vec{d}]\vec{c} - [\vec{a}\vec{b}\vec{c}]\vec{d}\]
\[\phantom{(\vec{a}\times\vec{b})\times(\vec{c}\times\vec{d})} = [\vec{a}\vec{c}\vec{d}]\vec{b} - [\vec{b}\vec{c}\vec{d}]\vec{a}\]
Express cross product of two cross products in terms of box products; linear combinations of original vectors.
Linear Dependence of 4 Vectors
\[[\vec{a}\vec{b}\vec{d}]\vec{c} - [\vec{a}\vec{b}\vec{c}]\vec{d} = [\vec{a}\vec{c}\vec{d}]\vec{b} - [\vec{b}\vec{c}\vec{d}]\vec{a}\]
Any 4 vectors in 3D — if no 3 are coplanar — can express one as linear combo of the other 3.
◈ Special Geometry (Tetrahedron, Hexagon, etc.) 5 ▶
Regular Hexagon — Diagonal Sum
\[\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}+\overrightarrow{OE}+\overrightarrow{OF} = \vec{0}\]
Opposite vertices of regular hexagon give equal and opposite vectors; their sum is zero.
Tetrahedron Centre
\[\vec{g} = \frac{\vec{a}+\vec{b}+\vec{c}+\vec{d}}{4}\]
Point where vertex-to-opposite-centroid lines concur; lines joining midpoints of opposite edges also concur here.
Parallelepiped Centre
\[\text{Centre} = \frac{\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}}{2}\]
Intersection point of 4 diagonals of a parallelepiped; they bisect each other.
Parallelepiped = Tetrahedron Centre
\[\text{Centre of } \|^{piped} = \text{Centre of inscribed tetrahedron}\]
Distance between centres of the parallelepiped ABCDEFGH and tetrahedron FAHC is zero.
Solving Vector Equations
\[\vec{r}\times\vec{b} = \vec{a} \Rightarrow \vec{r} = y\vec{b} - \frac{\vec{a}\times\vec{b}}{|\vec{b}|^2}\]
\[\vec{A}\cdot\vec{x} = c,\ \vec{A}\times\vec{x} = \vec{B} \Rightarrow \vec{x} = \frac{c\vec{A}-(\vec{A}\times\vec{B})}{|\vec{A}|^2}\]
Isolate unknown vector when both dot and cross product constraints are given.
📐 3D Geometry 10 SECTIONS
◈ Distance, Section & Centroid 7 ▶
Distance Between Two Points
\[AB = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\]
Foundation of all 3D geometry; test collinearity, find side lengths, check right angles.
Distance from Origin
\[OP = \sqrt{x^2+y^2+z^2}\]
Find distance of any point P(x,y,z) from the origin.
Distance from Coordinate Axes
\[PA = \sqrt{y^2+z^2},\quad PB = \sqrt{z^2+x^2},\quad PC = \sqrt{x^2+y^2}\]
Shortest distances from P to x-axis (PA), y-axis (PB), z-axis (PC) respectively.
Section Formula — Internal
\[R = \left(\frac{m_1x_2+m_2x_1}{m_1+m_2},\ \frac{m_1y_2+m_2y_1}{m_1+m_2},\ \frac{m_1z_2+m_2z_1}{m_1+m_2}\right)\]
Point dividing PQ internally in ratio m₁:m₂; use to find where a plane cuts a line.
Midpoint
\[M = \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)\]
Centre of an edge; circumcenter of a right-angled triangle is midpoint of hypotenuse.
Centroid of Triangle
\[G = \left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3},\frac{z_1+z_2+z_3}{3}\right)\]
Centroid divides each median in 2:1; equidistant from vertices in equilateral triangle.
Centroid of Tetrahedron
\[G = \left(\frac{\Sigma x_i}{4},\frac{\Sigma y_i}{4},\frac{\Sigma z_i}{4}\right)\]
Point where vertex-to-opposite-face-centroid lines concur; equidistant from all faces in regular tetrahedron.
◈ Direction Cosines & Ratios 10 ▶
Direction Cosines (DC's) Definition
\[\ell = \cos\alpha,\quad m=\cos\beta,\quad n=\cos\gamma\]
α,β,γ = angles line makes with x,y,z axes; DC's uniquely characterise direction (up to sign).
Fundamental Identity
\[\ell^2+m^2+n^2 = 1 \iff \cos^2\alpha+\cos^2\beta+\cos^2\gamma = 1\]
\[\sin^2\alpha+\sin^2\beta+\sin^2\gamma = 2\]
Always use to find missing DC; sum of squares always equals 1. Sin version = 3 − 1 = 2.
DC's of Coordinate Axes
\[x\text{-axis}: (1,0,0),\quad y\text{-axis}: (0,1,0),\quad z\text{-axis}: (0,0,1)\]
Reference DC's; used to find angles a line makes with each axis.
Relation: DC's from DR's
\[\ell = \frac{\pm a}{\sqrt{a^2+b^2+c^2}},\quad m = \frac{\pm b}{\sqrt{a^2+b^2+c^2}},\quad n = \frac{\pm c}{\sqrt{a^2+b^2+c^2}}\]
Convert direction ratios a,b,c to actual direction cosines (normalise by magnitude).
DR's of Line Joining Two Points
\[a = x_2-x_1,\quad b = y_2-y_1,\quad c = z_2-z_1\]
Direction ratios of AB; directly usable in symmetric line equation.
Point on Line at Distance r from (x₁,y₁,z₁)
\[P = (x_1+\ell r,\ y_1+mr,\ z_1+nr)\]
\[\ell,m,n \text{ must be DC's (not DR's) for } AP=r\]
Find specific points on a line at given distance; only works with actual DC's.
Projection of Line Segment on Another Line
\[\text{Proj of }AB \text{ on line with DC's }(\ell,m,n):\]
\[= |\ell(x_2-x_1)+m(y_2-y_1)+n(z_2-z_1)|\]
Length of shadow of AB onto a given line; used in distance-related problems.
DC's of Line ⊥ to Two Given Lines
\[\frac{\ell}{m_1n_2-m_2n_1} = \frac{m}{n_1\ell_2-n_2\ell_1} = \frac{n}{\ell_1m_2-\ell_2m_1}\]
DC's of the cross product of two direction vectors; normal to the plane containing them.
Angle with 4 Diagonals of Cube
\[\cos^2\alpha+\cos^2\beta+\cos^2\gamma+\cos^2\delta = \frac{4}{3}\]
A line making angles α,β,γ,δ with the four space diagonals of a unit cube satisfies this.
DC's Uniqueness
\[\text{A line has exactly 2 sets of DC's: } (\ell,m,n) \text{ and } (-\ell,-m,-n)\]
\[\text{DR's are infinite (any scalar multiples)}\]
Direction ratios are not unique; DC's are unique up to overall sign reversal.
◈ Equations of a Plane 9 ▶
General Equation
\[ax + by + cz + d = 0\]
Normal vector is (a,b,c); every first-degree equation in x,y,z represents a plane.
Vector Form
\[(\vec{r}-\vec{a})\cdot\vec{n} = 0 \implies \vec{r}\cdot\vec{n} = d, \quad d = \vec{a}\cdot\vec{n}\]
Plane through point a⃗ with normal n⃗; compact form for distance and angle calculations.
Plane Through a Given Point
\[a(x-x_1)+b(y-y_1)+c(z-z_1) = 0\]
Write plane equation when a point on it and its normal direction are known.
Coordinate Planes & Parallel Planes
\[xy\text{-plane}: z=0,\quad yz\text{-plane}: x=0,\quad zx\text{-plane}: y=0\]
\[\text{Parallel to }xy: z=c,\quad \text{to }yz: x=c,\quad \text{to }zx: y=c\]
Find where a plane cuts axes; write equation of plane parallel to a coordinate plane.
Planes Parallel to an Axis
\[\text{Parallel to }x: by+cz+d=0\]
\[\text{Parallel to }y: ax+cz+d=0\]
\[\text{Parallel to }z: ax+by+d=0\]
When plane contains one axis direction; coefficient of that variable is zero.
Intercept Form
\[\frac{x}{a}+\frac{y}{b}+\frac{z}{c} = 1\]
Plane cutting axes at A(a,0,0), B(0,b,0), C(0,0,c); useful for centroid/area problems.
Normal Form (Cartesian)
\[\ell x + my + nz = p\]
ℓ,m,n are DC's of normal from origin; p = perpendicular distance from origin.
Normal Form (Vector)
\[\vec{r}\cdot\hat{n} = d\]
n̂ = unit normal; d = signed distance from origin; immediate perpendicular-distance form.
Plane Through Three Points
\[\begin{vmatrix}x-x_1 & y-y_1 & z-z_1\\x_2-x_1 & y_2-y_1 & z_2-z_1\\x_3-x_1 & y_3-y_1 & z_3-z_1\end{vmatrix} = 0\]
Unique plane through three non-collinear points; expand the 3×3 determinant.
◈ Distance, Angle & Bisectors of Planes 9 ▶
Perpendicular Distance from Point to Plane
\[p = \frac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}}\]
Most-used formula; plug coordinates into plane equation and normalise.
Vector Distance from Point to Plane
\[p = \frac{|\vec{a}\cdot\hat{n}-d|}{|\vec{n}|}\]
Vector form of perpendicular distance; useful when plane is given in r⃗·n⃗=d form.
Distance Between Parallel Planes
\[d = \frac{|d_1-d_2|}{\sqrt{a^2+b^2+c^2}}\]
Distance between ax+by+cz+d₁=0 and ax+by+cz+d₂=0 (same normal, different constants).
Angle Between Two Planes
\[\cos\theta = \frac{|aa'+bb'+cc'|}{\sqrt{a^2+b^2+c^2}\cdot\sqrt{a'^2+b'^2+c'^2}}\]
Angle between planes = angle between their normals; absolute value gives acute angle.
Perpendicular Planes
\[aa'+bb'+cc' = 0\]
Dot product of normals is zero iff planes are perpendicular.
Parallel Planes
\[\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'}\]
Normals are proportional iff planes are parallel; to find parallel plane, change only constant d.
Angle Bisector Planes
\[\frac{ax+by+cz+d}{\sqrt{a^2+b^2+c^2}} = \pm\frac{a_1x+b_1y+c_1z+d_1}{\sqrt{a_1^2+b_1^2+c_1^2}}\]
Locus of points equidistant from two planes; gives two bisector planes (+ and −).
Acute vs Obtuse Bisector (Origin Test)
\[aa_1+bb_1+cc_1 > 0 \Rightarrow \text{origin in obtuse angle (use − sign for acute bisector)}\]
\[aa_1+bb_1+cc_1 < 0 \Rightarrow \text{origin in acute angle (use + sign for acute bisector)}\]
Identify which bisector plane is for the acute and which for the obtuse angle.
Family of Planes (Through Line of Intersection)
\[P_1 + \lambda P_2 = 0\]
\[(a_1x+b_1y+c_1z+d_1) + \lambda(a_2x+b_2y+c_2z+d_2) = 0\]
All planes through the line of intersection of P₁=0 and P₂=0; find λ using extra condition.
◈ Equations of Straight Lines 7 ▶
Symmetric / Parametric Form
\[\frac{x-x_1}{\ell} = \frac{y-y_1}{m} = \frac{z-z_1}{n} = r\]
Standard form; ℓ,m,n are DC's (or DR's); r gives distance along line if DC's used.
Any Point on the Line at Parameter r
\[P = (x_1+\ell r,\ y_1+mr,\ z_1+nr)\]
Substitute into plane equations to find intersection point of line and plane.
Line Through Two Points
\[\frac{x-x_1}{x_2-x_1} = \frac{y-y_1}{y_2-y_1} = \frac{z-z_1}{z_2-z_1}\]
Passes through A(x₁,y₁,z₁) and B(x₂,y₂,z₂); DR's are differences of coordinates.
Unsymmetrical Form
\[a_1x+b_1y+c_1z+d_1=0\quad\text{AND}\quad a_2x+b_2y+c_2z+d_2=0\]
Intersection of two non-parallel planes defines a line; convert to symmetric form when needed.
Converting Unsymmetric → Symmetric
\[\text{Step 1: DR's } \vec{b} = \vec{n}_1\times\vec{n}_2\]
\[\text{Step 2: Find point (set }z=0\text{, solve 2×2 system)}\]
\[\text{Step 3: Write symmetric form through that point with DR's }\vec{b}\]
Convert intersection of two planes to a usable line equation.
Particular Lines
\[x\text{-axis}: \frac{x}{1}=\frac{y}{0}=\frac{z}{0},\quad y\text{-axis}: \frac{x}{0}=\frac{y}{1}=\frac{z}{0},\quad z\text{-axis}: \frac{x}{0}=\frac{y}{0}=\frac{z}{1}\]
\[\parallel x: y=p, z=q;\quad \parallel y: x=h, z=q;\quad \parallel z: x=h, y=p\]
Reference forms for standard axes and lines parallel to them.
3 Planes — System Classification
\[\text{Unique solution} \Rightarrow \text{lines intersect at a point}\]
\[\text{Infinite solutions} \Rightarrow \text{planes intersect coaxially}\]
\[\text{No solution} \Rightarrow \text{no common point}\]
Use Cramer's rule or row reduction to classify three-plane system.
◈ Line–Line & Line–Plane Angles 8 ▶
Angle Between Two Lines
\[\cos\theta = |\ell_1\ell_2+m_1m_2+n_1n_2|\]
\[\cos\theta = \frac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\cdot\sqrt{a_2^2+b_2^2+c_2^2}}\]
Angle between directions of two lines in 3D; gives acute angle (absolute value).
Perpendicular Lines
\[\ell_1\ell_2+m_1m_2+n_1n_2=0 \iff a_1a_2+b_1b_2+c_1c_2=0\]
Dot product of direction vectors is zero; test for perpendicular lines or faces.
Parallel Lines
\[\frac{\ell_1}{\ell_2}=\frac{m_1}{m_2}=\frac{n_1}{n_2} \iff \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\]
Direction ratios proportional; used to write parallel line through a given point.
Angle Between Line and Plane
\[\sin\theta = \frac{|a\ell+bm+cn|}{\sqrt{a^2+b^2+c^2}\cdot\sqrt{\ell^2+m^2+n^2}}\]
Complement of angle between line and normal to plane; sin (not cos) because line is in plane when θ=0.
Line Parallel to Plane
\[\theta=0 \iff a\ell+bm+cn = 0\]
Direction of line perpendicular to normal of plane; line does not intersect plane.
Line Perpendicular to Plane
\[\frac{a}{\ell}=\frac{b}{m}=\frac{c}{n}\]
Line direction proportional to plane normal; the line is perpendicular (passes through plane).
Conditions for Line to Lie in Plane
\[\text{(i) } A\ell+Bm+Cn = 0 \quad\text{(direction }\perp\text{ normal)}\]
\[\text{(ii) } Ax_1+By_1+Cz_1+D = 0 \quad\text{(point on plane)}\]
Both conditions must hold; used to find unknown constants in mixed problems.
Position of Two Points w.r.t. Plane
\[\text{Same side} \iff (ax_1+by_1+cz_1+d)(ax_2+by_2+cz_2+d) > 0\]
\[\text{Opposite sides} \iff \text{product} < 0\]
Test whether two points lie on the same or opposite sides of a given plane.
◈ Foot, Image & Perpendicular 6 ▶
Foot of Perpendicular from Point to Line
\[\text{Line: } \frac{x-x_1}{\ell}=\frac{y-y_1}{m}=\frac{z-z_1}{n}\]
\[r = (\alpha-x_1)\ell+(\beta-y_1)m+(\gamma-z_1)n\]
\[\text{Foot } P = (x_1+\ell r,\ y_1+mr,\ z_1+nr)\]
Find the closest point on a line to a given external point A(α,β,γ).
Length of Perpendicular from Point to Line
\[AP = \sqrt{(\ell r+x_1-\alpha)^2+(mr+y_1-\beta)^2+(nr+z_1-\gamma)^2}\]
Shortest distance from external point A to a line; substitute foot coordinates.
Image of Point P in a Plane
\[\frac{x_2-x_1}{a}=\frac{y_2-y_1}{b}=\frac{z_2-z_1}{c} = \frac{-2(ax_1+by_1+cz_1+d)}{a^2+b^2+c^2}\]
Reflect point (x₁,y₁,z₁) in the plane ax+by+cz+d=0 to get image (x₂,y₂,z₂).
Image Method: Step by Step
\[\text{1. Line PQ: }\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}=r\]
\[\text{2. Midpoint R of PQ lies on plane: substitute mid-coords}\]
\[\text{3. Solve for }r, \text{ then } Q=(x_1+2ar,\ y_1+2br,\ z_1+2cr)\]
Systematic approach to find image when the formula is not memorised.
Foot of Perpendicular from Point to Plane
\[\text{Foot} = (x_1-a\lambda,\ y_1-b\lambda,\ z_1-c\lambda)\]
\[\lambda = \frac{ax_1+by_1+cz_1+d}{a^2+b^2+c^2}\]
Closest point on plane to P(x₁,y₁,z₁); foot lies on perpendicular from P to plane.
Equation of Plane Through a Point ⊥ to OP
\[hx+ky+\ell z = h^2+k^2+\ell^2 = p^2\]
Plane perpendicular to OP at P(h,k,ℓ) from origin; normal form with p=|OP|.
◈ Coplanarity & Shortest Distance (Lines) 6 ▶
Coplanarity of Two Lines (Determinant)
\[\begin{vmatrix}x_1-x_2 & y_1-y_2 & z_1-z_2\\\ell_1 & m_1 & n_1\\\ell_2 & m_2 & n_2\end{vmatrix} = 0\]
Lines L₁ and L₂ are coplanar (intersecting or parallel) iff this determinant is zero.
Equation of Plane Containing Two Coplanar Lines
\[\begin{vmatrix}x-x_1 & y-y_1 & z-z_1\\\ell_1 & m_1 & n_1\\\ell_2 & m_2 & n_2\end{vmatrix} = 0\]
Write equation of plane containing both lines once coplanarity is confirmed.
SD Between Skew Lines (Cartesian)
\[\text{SD} = \frac{\left|\begin{vmatrix}x_2-x_1 & y_2-y_1 & z_2-z_1\\\ell_1 & m_1 & n_1\\\ell_2 & m_2 & n_2\end{vmatrix}\right|}{\sqrt{\Sigma(m_1n_2-m_2n_1)^2}}\]
Shortest distance in component form; denominator = |b⃗₁×b⃗₂|.
SD Between Skew Lines (Vector)
\[\text{SD} = \frac{|(\vec{a}_2-\vec{a}_1)\cdot(\vec{b}_1\times\vec{b}_2)|}{|\vec{b}_1\times\vec{b}_2|}\]
Compact vector form; project the connecting vector onto the common perpendicular direction.
SD Between Parallel Lines
\[d = \frac{|\vec{b}\times(\vec{a}_2-\vec{a}_1)|}{|\vec{b}|}\]
Both lines have same direction b⃗; distance is perpendicular separation between them.
Coplanarity of 4 Points
\[[\overrightarrow{AB}\ \overrightarrow{AC}\ \overrightarrow{AD}] = 0\]
\[\iff \begin{vmatrix}x_2-x_1&y_2-y_1&z_2-z_1\\x_3-x_1&y_3-y_1&z_3-z_1\\x_4-x_1&y_4-y_1&z_4-z_1\end{vmatrix}=0\]
Test whether four points lie in a single plane; box product of three edge vectors is zero.
◈ Area of Triangle & Special Formulas 6 ▶
Area of Triangle (Cross Product)
\[\Delta = \frac{1}{2}|\overrightarrow{AB}\times\overrightarrow{AC}|\]
Direct formula using two edge vectors from same vertex; most efficient method.
Area via Projections
\[\Delta_x = \frac{1}{2}\begin{vmatrix}y_1&z_1&1\\y_2&z_2&1\\y_3&z_3&1\end{vmatrix},\quad \Delta_y = \frac{1}{2}\begin{vmatrix}x_1&z_1&1\\x_2&z_2&1\\x_3&z_3&1\end{vmatrix},\quad \Delta_z = \frac{1}{2}\begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{vmatrix}\]
\[\Delta = \sqrt{\Delta_x^2+\Delta_y^2+\Delta_z^2}\]
Area from coordinate projections; Δx, Δy, Δz are areas of triangle's projections on coordinate planes.
Line of Greatest Slope
\[\text{Line in G-plane, } \perp \text{ to } (G\cap H)\text{-line, through a given point}\]
Steepest line on a given plane (G-plane) relative to horizontal plane H; find by: direction ⊥ to line of intersection.
Length PQ Between Two Parallel Planes
\[\text{Line with DR's } (a,b,c) \text{ cuts planes } P_1, P_2\text{ at angle }\theta:\]
\[PQ = \frac{\text{dist between planes}}{\sin\theta} = \frac{|d_1-d_2|}{\sqrt{a^2+b^2+c^2}\cdot\sin\theta}\]
Length intercepted on an oblique line between two parallel planes.
Locus: Sum of Distances from Two Points = k
\[\text{PA}+\text{PB}=k \Rightarrow \text{Ellipsoid (degenerate cases: sphere or plane)}\]
\[\text{Equal dist from two pts} \Rightarrow \text{a plane (perpendicular bisector plane)}\]
Locus problems in 3D; equal distance gives plane, sum of distances gives ellipsoid.
Variable Plane at Constant Distance p
\[\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2} = \frac{1}{p^2}\]
\[\text{Locus of centroid of }\triangle ABC: \frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2} = \frac{9}{p^2}\]
Plane at constant distance p from origin cutting axes at A,B,C; locus of centroid of △ABC.
◈ Vector-Form 3D (Quick Reference) 6 ▶
Vector Along Line of Intersection of Two Planes
\[\vec{b} = \vec{n}_1 \times \vec{n}_2\]
Direction of the line formed by intersecting planes with normals n⃗₁ and n⃗₂.
Plane Containing Line of Intersection of P₁, P₂
\[P_1 + \lambda P_2 = 0\]
\[(\vec{r}\cdot\vec{n}_1-d_1)+\lambda(\vec{r}\cdot\vec{n}_2-d_2)=0\]
One-parameter family; substitute extra condition (passes through a point, ⊥ to a vector, etc.) to find λ.
Distance from Origin to Plane r·n̂ = d
\[p = \frac{|d|}{|\vec{n}|} = d \quad \text{(if }\hat{n}\text{ is unit vector)}\]
When plane is given in normal form, d directly equals perpendicular distance from origin.
Angle Between Line r=a+tb and Plane r·n=d
\[\sin\theta = \frac{|\vec{b}\cdot\vec{n}|}{|\vec{b}||\vec{n}|}\]
Vector-form angle formula; b⃗ = direction of line, n⃗ = normal to plane.
Angle Between Planes r·n₁=d₁ and r·n₂=d₂
\[\cos\theta = \frac{|\vec{n}_1\cdot\vec{n}_2|}{|\vec{n}_1||\vec{n}_2|}\]
Vector form; n₁·n₂=0 for perpendicular planes; n₁∥n₂ for parallel planes.
Foot of ⊥ from a⃗ to Plane r·n̂=d
\[\text{Foot} = \vec{a} - (\vec{a}\cdot\hat{n}-d)\hat{n}\]
\[\text{Image} = \vec{a} - 2(\vec{a}\cdot\hat{n}-d)\hat{n}\]
Vector formula for foot and image of a point in a plane given in normal form.