Vector Algebra

A physical quantity having only magnitude (size or number) is termed a scalar. Examples include mass, time, temperature, and arc length. In contrast, a quantity possessing both magnitude and a specific direction which obeys the parallelogram law of addition is termed a vector. Examples include displacement, velocity, acceleration, force, and torque.

A vector is typically denoted by a bold lowercase letter $\mathbf{a}$ or by directed segment notation $\overrightarrow{AB}$, where $A$ is the initial point (tail) and $B$ is the terminal point (head). The magnitude of vector $\mathbf{a}$ is denoted $|\mathbf{a}|$ and represents the length of the directed segment.

• Zero (Null) Vector $\mathbf{0}$: A vector with zero magnitude and an indeterminate direction. The starting and terminal points coincide.
• Unit Vector $\hat{\mathbf{a}}$: A vector of magnitude unity in the direction of $\mathbf{a}$, defined as $\hat{\mathbf{a}} = \dfrac{\mathbf{a}}{|\mathbf{a}|}$, valid only when $|\mathbf{a}| \neq 0$.
• Equal Vectors: Two vectors are equal if they have the same magnitude and the same direction, regardless of initial point positions.
• Negative Vector: The vector $-\mathbf{a}$ has the same magnitude as $\mathbf{a}$ but opposite direction.
• Collinear (Parallel) Vectors: Vectors lying along the same line or parallel lines, irrespective of magnitude and sense. They satisfy $\mathbf{a} = \lambda \mathbf{b}$ for some scalar $\lambda \in \mathbb{R}$.
• Co-initial Vectors: Vectors sharing the same initial point.
• Coplanar Vectors: Three or more vectors lying in or parallel to the same plane.
• Free Vector: A vector whose position in space is arbitrary; only magnitude and direction matter.
• Localised Vector: A vector with a fixed initial point (used in mechanics for forces applied at specific points).

The position vector of a point $P(x, y, z)$ relative to the origin $O$ is $\overrightarrow{OP} = x\hat{i} + y\hat{j} + z\hat{k}$, with magnitude $|\overrightarrow{OP}| = \sqrt{x^2 + y^2 + z^2}$. Here, $\hat{i}$, $\hat{j}$, $\hat{k}$ are mutually perpendicular unit vectors along the positive $x$, $y$, $z$ axes respectively.

• Triangle Law: If the tail of $\mathbf{b}$ is placed at the head of $\mathbf{a}$, then $\mathbf{a} + \mathbf{b}$ is the vector from the tail of $\mathbf{a}$ to the head of $\mathbf{b}$.
• Parallelogram Law: If $\mathbf{a}$ and $\mathbf{b}$ are co-initial, then their sum is the diagonal of the parallelogram formed by $\mathbf{a}$ and $\mathbf{b}$.

• Commutativity: $\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}$
• Associativity: $(\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c})$
• Existence of identity: $\mathbf{a} + \mathbf{0} = \mathbf{a}$
• Existence of inverse: $\mathbf{a} + (-\mathbf{a}) = \mathbf{0}$

Section Formula: If point $P$ divides the line segment joining points $A$ (position vector $\mathbf{a}$) and $B$ (position vector $\mathbf{b}$) in the ratio $m : n$, then the position vector of $P$ is: \[ \mathbf{r}_P = \frac{m\mathbf{b} + n\mathbf{a}}{m + n} \quad \text{(internal division)} \] \[ \mathbf{r}_P = \frac{m\mathbf{b} - n\mathbf{a}}{m - n} \quad \text{(external division)} \] The midpoint of $AB$ has position vector $\mathbf{r}_M = \dfrac{\mathbf{a} + \mathbf{b}}{2}$.

Linear Combination and Linear Independence: A vector $\mathbf{r}$ is a linear combination of vectors $\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n$ if there exist scalars $\lambda_1, \lambda_2, \ldots, \lambda_n$ such that $\mathbf{r} = \lambda_1 \mathbf{a}_1 + \lambda_2 \mathbf{a}_2 + \cdots + \lambda_n \mathbf{a}_n$.

• Three points $A$, $B$, $C$ with position vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are collinear iff $\overrightarrow{AB} = \lambda \overrightarrow{AC}$, equivalently $\exists$ scalars $x, y, z$ (not all zero) such that $x\mathbf{a} + y\mathbf{b} + z\mathbf{c} = \mathbf{0}$ with $x + y + z = 0$.
• Four points $A, B, C, D$ are coplanar iff $\exists$ scalars $x, y, z, w$ (not all zero) such that $x\mathbf{a} + y\mathbf{b} + z\mathbf{c} + w\mathbf{d} = \mathbf{0}$ and $x + y + z + w = 0$.

The scalar product (dot product) of two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$, inclined at angle $\theta$ ($0 \leq \theta \leq \pi$), is the scalar: \[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta \] If $\mathbf{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\mathbf{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, then in component form: \[ \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 \]

• Commutativity: $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$
• Distributivity: $\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}$
• Scalar multiplication: $(\lambda \mathbf{a}) \cdot \mathbf{b} = \lambda(\mathbf{a} \cdot \mathbf{b})$
• $\mathbf{a} \cdot \mathbf{a} = |\mathbf{a}|^2$
• Perpendicularity: $\mathbf{a} \perp \mathbf{b} \iff \mathbf{a} \cdot \mathbf{b} = 0$ (for non-zero vectors)
• For standard basis: $\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1$; $\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0$

Angle Between Two Vectors: \[ \cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|} \]

Projection (Component) of $\mathbf{a}$ along $\mathbf{b}$: The scalar projection (signed length) of $\mathbf{a}$ on $\mathbf{b}$ is: \[ \text{Scalar projection of } \mathbf{a} \text{ on } \mathbf{b} = |\mathbf{a}|\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|} \] The vector projection of $\mathbf{a}$ on $\mathbf{b}$ is: \[ \text{proj}_\mathbf{b}\mathbf{a} = \left(\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|^2}\right)\mathbf{b} = \left(\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|}\right)\hat{\mathbf{b}} \]

Direction Cosines: If $\mathbf{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$, then the direction cosines (with respect to coordinate axes) are: \[ \ell = \frac{a_1}{|\mathbf{a}|}, \quad m = \frac{a_2}{|\mathbf{a}|}, \quad n = \frac{a_3}{|\mathbf{a}|}, \quad \text{and} \quad \ell^2 + m^2 + n^2 = 1 \]

Useful Identity (Polarization): \[ |\mathbf{a} + \mathbf{b}|^2 = |\mathbf{a}|^2 + |\mathbf{b}|^2 + 2(\mathbf{a} \cdot \mathbf{b}) \] \[ |\mathbf{a} - \mathbf{b}|^2 = |\mathbf{a}|^2 + |\mathbf{b}|^2 - 2(\mathbf{a} \cdot \mathbf{b}) \] \[ |\mathbf{a} + \mathbf{b}|^2 + |\mathbf{a} - \mathbf{b}|^2 = 2(|\mathbf{a}|^2 + |\mathbf{b}|^2) \quad \text{(Parallelogram identity)} \]

The dot product offers an elegant route to several classical results in geometry:

Cosine Rule (Vector Proof): In triangle $ABC$ with $\overrightarrow{BC} = \mathbf{a}$, $\overrightarrow{CA} = \mathbf{b}$, $\overrightarrow{AB} = \mathbf{c}$, the relation $\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}$ yields: \[ a^2 = b^2 + c^2 - 2bc \cos A \]

• Equal in length iff $|\mathbf{a} + \mathbf{b}| = |\mathbf{a} - \mathbf{b}|$ iff $\mathbf{a} \cdot \mathbf{b} = 0$ (rectangle)
• Perpendicular iff $|\mathbf{a}| = |\mathbf{b}|$ (rhombus)

Equation of a Line: The vector equation of a line passing through point with position vector $\mathbf{a}$ and parallel to vector $\mathbf{b}$ is: \[ \mathbf{r} = \mathbf{a} + \lambda \mathbf{b}, \quad \lambda \in \mathbb{R} \] The line through two points $\mathbf{a}$ and $\mathbf{b}$ has equation: \[ \mathbf{r} = \mathbf{a} + \lambda(\mathbf{b} - \mathbf{a}) = (1 - \lambda)\mathbf{a} + \lambda \mathbf{b} \]

Component Resolution: Any vector $\mathbf{a}$ can be decomposed uniquely into components parallel and perpendicular to a unit vector $\hat{\mathbf{b}}$: \[ \mathbf{a} = (\mathbf{a} \cdot \hat{\mathbf{b}})\hat{\mathbf{b}} + \left[\mathbf{a} - (\mathbf{a} \cdot \hat{\mathbf{b}})\hat{\mathbf{b}}\right] \] The first term is parallel to $\mathbf{b}$, the bracket is perpendicular to $\mathbf{b}$.

Work Done by a Force: If a constant force $\mathbf{F}$ produces displacement $\mathbf{d}$, the work done is the scalar: \[ W = \mathbf{F} \cdot \mathbf{d} = |\mathbf{F}||\mathbf{d}|\cos\theta \]

The vector product (cross product) of two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ at angle $\theta$ ($0 \leq \theta \leq \pi$) is defined as the vector: \[ \mathbf{a} \times \mathbf{b} = |\mathbf{a}||\mathbf{b}|\sin\theta \, \hat{\mathbf{n}} \] where $\hat{\mathbf{n}}$ is the unit vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$ such that $\{\mathbf{a}, \mathbf{b}, \hat{\mathbf{n}}\}$ forms a right-handed system.

Determinant Form (Coordinate Definition): If $\mathbf{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\mathbf{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, then: \[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} = (a_2 b_3 - a_3 b_2)\hat{i} - (a_1 b_3 - a_3 b_1)\hat{j} + (a_1 b_2 - a_2 b_1)\hat{k} \]

• Anti-commutativity: $\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})$
• Distributivity: $\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}$
• $\mathbf{a} \times \mathbf{a} = \mathbf{0}$
• Parallelism: $\mathbf{a} \parallel \mathbf{b} \iff \mathbf{a} \times \mathbf{b} = \mathbf{0}$
• Standard basis: $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, $\hat{k} \times \hat{i} = \hat{j}$ (cyclic)
• Lagrange's Identity: $|\mathbf{a} \times \mathbf{b}|^2 = |\mathbf{a}|^2|\mathbf{b}|^2 - (\mathbf{a} \cdot \mathbf{b})^2$

• $|\mathbf{a} \times \mathbf{b}|$ = area of parallelogram with adjacent sides $\mathbf{a}$ and $\mathbf{b}$
• Area of triangle with adjacent sides $\mathbf{a}, \mathbf{b}$ = $\dfrac{1}{2}|\mathbf{a} \times \mathbf{b}|$
• Area of triangle with vertices $A, B, C$ = $\dfrac{1}{2}|\overrightarrow{AB} \times \overrightarrow{AC}|$
• Unit vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$: $\hat{\mathbf{n}} = \pm \dfrac{\mathbf{a} \times \mathbf{b}}{|\mathbf{a} \times \mathbf{b}|}$

• Moment of force $\mathbf{F}$ about point $O$ with position vector $\mathbf{r}$: $\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}$
• Velocity of a point in rotation: $\mathbf{v} = \boldsymbol{\omega} \times \mathbf{r}$

The scalar triple product (STP) of three vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ is the scalar: \[ [\mathbf{a}\,\mathbf{b}\,\mathbf{c}] = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} \] In component form, if $\mathbf{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$, etc.: \[ [\mathbf{a}\,\mathbf{b}\,\mathbf{c}] = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} \]

• Cyclic property: $[\mathbf{a}\,\mathbf{b}\,\mathbf{c}] = [\mathbf{b}\,\mathbf{c}\,\mathbf{a}] = [\mathbf{c}\,\mathbf{a}\,\mathbf{b}]$
• Sign change with interchange: $[\mathbf{a}\,\mathbf{b}\,\mathbf{c}] = -[\mathbf{b}\,\mathbf{a}\,\mathbf{c}]$
• Coplanarity test: $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are coplanar iff $[\mathbf{a}\,\mathbf{b}\,\mathbf{c}] = 0$
• If any two of $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are equal or parallel, then $[\mathbf{a}\,\mathbf{b}\,\mathbf{c}] = 0$
• Scalar multiplication: $[\lambda \mathbf{a}\,\mathbf{b}\,\mathbf{c}] = \lambda [\mathbf{a}\,\mathbf{b}\,\mathbf{c}]$
• $[\mathbf{a} + \mathbf{d}\,\mathbf{b}\,\mathbf{c}] = [\mathbf{a}\,\mathbf{b}\,\mathbf{c}] + [\mathbf{d}\,\mathbf{b}\,\mathbf{c}]$

• Volume of tetrahedron with concurrent edges $\mathbf{a}, \mathbf{b}, \mathbf{c}$ from one vertex: $V = \dfrac{1}{6}|[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]|$
• Four points $A, B, C, D$ with position vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}$ are coplanar iff $[\overrightarrow{AB}\,\overrightarrow{AC}\,\overrightarrow{AD}] = 0$

Vector Triple Product (BAC--CAB Rule): \[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c} \] \[ (\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{b} \cdot \mathbf{c})\mathbf{a} \] Note that the vector triple product is not associative; in general, $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \neq (\mathbf{a} \times \mathbf{b}) \times \mathbf{c}$.

Lagrange's Identity for STP: \[ [\mathbf{a} \times \mathbf{b}\,\,\mathbf{c} \times \mathbf{d}\,\,\mathbf{e} \times \mathbf{f}] \text{ (advanced expansion exists)} \] A more frequently used identity: \[ (\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{c} \times \mathbf{d}) = \begin{vmatrix} \mathbf{a}\cdot\mathbf{c} & \mathbf{a}\cdot\mathbf{d} \\ \mathbf{b}\cdot\mathbf{c} & \mathbf{b}\cdot\mathbf{d} \end{vmatrix} \]

A line in space is uniquely determined by a point and a direction vector. The vector equation of a line through a point with position vector $\mathbf{a}$ and parallel to $\mathbf{b}$ is: \[ \mathbf{r} = \mathbf{a} + \lambda \mathbf{b}, \quad \lambda \in \mathbb{R} \]

Distance from a Point to a Line: The perpendicular distance $d$ from a point $P$ with position vector $\mathbf{p}$ to the line $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$ is: \[ d = \frac{|(\mathbf{p} - \mathbf{a}) \times \mathbf{b}|}{|\mathbf{b}|} \]

Shortest Distance between Two Skew Lines: Consider two lines: \[ L_1: \mathbf{r} = \mathbf{a}_1 + \lambda \mathbf{b}_1, \qquad L_2: \mathbf{r} = \mathbf{a}_2 + \mu \mathbf{b}_2 \] These lines are skew (non-intersecting and non-parallel) when $\mathbf{b}_1 \times \mathbf{b}_2 \neq \mathbf{0}$ and the common perpendicular has non-zero length. The shortest distance between them is: \[ d = \frac{|[\mathbf{a}_2 - \mathbf{a}_1\,\,\mathbf{b}_1\,\,\mathbf{b}_2]|}{|\mathbf{b}_1 \times \mathbf{b}_2|} = \frac{|(\mathbf{a}_2 - \mathbf{a}_1) \cdot (\mathbf{b}_1 \times \mathbf{b}_2)|}{|\mathbf{b}_1 \times \mathbf{b}_2|} \]

If $\mathbf{b}_1 \times \mathbf{b}_2 = \mathbf{0}$ (the lines are parallel), the shortest distance is computed using the point-to-line formula.

Condition for Coplanarity of Two Lines: Two lines $L_1$ and $L_2$ are coplanar iff: \[ [\mathbf{a}_2 - \mathbf{a}_1\,\,\mathbf{b}_1\,\,\mathbf{b}_2] = 0 \]

• Plane through point $\mathbf{a}$ with normal $\mathbf{n}$: $\quad (\mathbf{r} - \mathbf{a}) \cdot \mathbf{n} = 0$, equivalently $\mathbf{r} \cdot \mathbf{n} = \mathbf{a} \cdot \mathbf{n} = d$
• Plane through three non-collinear points $\mathbf{a}, \mathbf{b}, \mathbf{c}$: $\quad (\mathbf{r} - \mathbf{a}) \cdot [(\mathbf{b} - \mathbf{a}) \times (\mathbf{c} - \mathbf{a})] = 0$
• Plane through point $\mathbf{a}$ parallel to vectors $\mathbf{b}$ and $\mathbf{c}$: $\quad \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} + \mu \mathbf{c}$

Distance from Point to Plane: The perpendicular distance from point $\mathbf{p}$ to the plane $\mathbf{r} \cdot \hat{\mathbf{n}} = d$ is: \[ \text{Distance} = |\mathbf{p} \cdot \hat{\mathbf{n}} - d| \]

Reciprocal System of Vectors (JEE Advanced): Given three non-coplanar vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$, their reciprocal system $\mathbf{a}', \mathbf{b}', \mathbf{c}'$ is defined by: \[ \mathbf{a}' = \frac{\mathbf{b} \times \mathbf{c}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}, \quad \mathbf{b}' = \frac{\mathbf{c} \times \mathbf{a}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}, \quad \mathbf{c}' = \frac{\mathbf{a} \times \mathbf{b}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]} \]

• Biorthogonality: $\mathbf{a} \cdot \mathbf{a}' = \mathbf{b} \cdot \mathbf{b}' = \mathbf{c} \cdot \mathbf{c}' = 1$, and all other cross dot products vanish.
• Reciprocal STP: $[\mathbf{a}'\,\mathbf{b}'\,\mathbf{c}'] = \dfrac{1}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}$
• The reciprocal of a reciprocal system gives the original system.
• Any vector $\mathbf{r}$ can be expanded as: \[ \mathbf{r} = (\mathbf{r} \cdot \mathbf{a}')\mathbf{a} + (\mathbf{r} \cdot \mathbf{b}')\mathbf{b} + (\mathbf{r} \cdot \mathbf{c}')\mathbf{c} \]

The reciprocal system is also called the dual basis and is fundamental in crystallography (used to describe reciprocal lattices).

Differentiation of Vector Functions: A vector-valued function $\mathbf{a}(t) = a_1(t)\hat{i} + a_2(t)\hat{j} + a_3(t)\hat{k}$ is differentiable if each component is differentiable. Then: \[ \frac{d\mathbf{a}}{dt} = \frac{da_1}{dt}\hat{i} + \frac{da_2}{dt}\hat{j} + \frac{da_3}{dt}\hat{k} \]

• Linearity: $\dfrac{d}{dt}[\mathbf{a} + \mathbf{b}] = \dfrac{d\mathbf{a}}{dt} + \dfrac{d\mathbf{b}}{dt}$
• Scalar product rule: $\dfrac{d}{dt}[\phi(t)\mathbf{a}] = \phi'(t)\mathbf{a} + \phi(t)\dfrac{d\mathbf{a}}{dt}$
• Dot product rule: $\dfrac{d}{dt}[\mathbf{a} \cdot \mathbf{b}] = \dfrac{d\mathbf{a}}{dt} \cdot \mathbf{b} + \mathbf{a} \cdot \dfrac{d\mathbf{b}}{dt}$
• Cross product rule: $\dfrac{d}{dt}[\mathbf{a} \times \mathbf{b}] = \dfrac{d\mathbf{a}}{dt} \times \mathbf{b} + \mathbf{a} \times \dfrac{d\mathbf{b}}{dt}$ (order matters!)
• STP rule: $\dfrac{d}{dt}[\mathbf{a}\,\mathbf{b}\,\mathbf{c}] = [\mathbf{a}'\,\mathbf{b}\,\mathbf{c}] + [\mathbf{a}\,\mathbf{b}'\,\mathbf{c}] + [\mathbf{a}\,\mathbf{b}\,\mathbf{c}']$

Important Theorem: If $|\mathbf{a}(t)| = $ constant, then $\mathbf{a}(t) \cdot \dfrac{d\mathbf{a}}{dt} = 0$. Hence, a vector of constant magnitude is perpendicular to its derivative. This explains why position vectors on a sphere always have radial-tangential decomposition.