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Vector Quantities in Mechanics

Vector quantities are physical observables possessing both magnitude and direction that obey the parallelogram law of vector addition. In classical mechanics, kinematics and dynamics are formulated using vectors to describe spatial translation, rotation, and forces acting on continuous media and point masses.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Definition of scalars and vectors, graphical representation, unit vectors, position vector, resolution of vectors in 2D and 3D Cartesian coordinates, dot and cross products, directional cosines, and basic applications to displacement, velocity, acceleration, and force.

PGT extension

Vector differential operators (gradient, divergence, curl), conservative fields, central forces, pseudo-vectors (axial vectors like angular momentum, torque, Coriolis force), coordinate transformations, and invariance under rotation.

Prerequisites

Elementary trigonometry and Cartesian geometry
Concept of scalar quantities and dimensions
Fundamental units and SI system

Classification of Physical Quantities

Scalars are completely specified by a magnitude with appropriate units, following ordinary algebraic laws (e.g., mass, time, temperature, work, energy).
Vectors require magnitude, direction, and specific spatial orientation, transforming under coordinate rotation according to linear transformation rules (e.g., displacement, velocity, force, momentum).
Tensors of rank zero are scalars, rank one are vectors, and higher ranks describe complex physical phenomena such as stress, strain, and moments of inertia.

Polar versus Axial (Pseudo) Vectors

Polar vectors represent linear translation and have a natural sense of direction associated with spatial displacement, reversing sign under spatial inversion (parity transformation r → -r). Examples include linear velocity, acceleration, and force.
Axial vectors (pseudo-vectors) represent rotational characteristics and are associated with a rotation axis. They do not invert their direction under spatial inversion (r → -r), retaining their sign because they are defined via cross products of two polar vectors.
Examples of axial vectors include angular momentum ($L = r \times p)$, torque ($\tau = r \times F)$, magnetic field, and the Coriolis force in rotating reference frames.

Algebraic Operations on Vectors

Vector addition is commutative (A + $B = B + A)$ and associative, performed graphically via the triangle or parallelogram law, or analytically via component-wise addition.
Scalar (dot) product yields a scalar quantity representing the projection of one vector onto another: A · $B = AB \cos(\theta)$, obeying distributive laws.
Vector (cross) product yields an axial vector perpendicular to the plane containing the operands: A × $B = AB \sin(\theta)\hat{n}$, obeying anti-commutativity (A × $B = -B \times A)$.

Differentiation and Integration of Vectors

A vector function r(t) depends on a scalar parameter such as time t, representing a curve in space traced by the tip of the vector.
The time derivative of a vector function yields its rate of change, evaluated by differentiating its Cartesian components: dr/d$t = (dx/dt)\hat{i} + (dy/dt)\hat{j} + (dz/dt)\hat{k}$.
Product rules apply to vector derivatives, such as d/dt(A · B) = (dA/dt) · B + A · (dB/dt) and d/dt(A × B) = (dA/dt) × B + A × (dB/dt).

Derivations

Magnitude and Direction of Resultant Vector via Parallelogram Law

Conditions: Two vectors P and Q acting at an angle θ relative to each other.

Construct a parallelogram with adjacent sides representing vectors P and Q, with the angle between their tails equal to θ.
Drop a perpendicular from the head of vector Q to the extended line of vector P, forming a right-angled triangle.
Using Pythagoras theorem on the large right-angled triangle, the square of the resultant magnitude R$^{2} = (P + Q \cos(\theta))^{2} + (Q \sin(\theta))^{2}$.
Expand the binomial: R$^{2} = P^{2} + 2PQ \cos(\theta) + Q^{2} \cos^{2}(\theta) + Q^{2} \sin^{2}(\theta) = P^{2} + Q^{2} + 2PQ \cos(\theta)$.
For direction angle α made by R with vector P, use trigonometry: tan(α) = (Q sin(θ)) / (P + Q cos(θ)).

Result: $R = \sqrt{P^2 + Q^2 + 2PQ \cos(\theta)}$, \quad \tan(\alpha) = \frac{Q \sin(\theta)}{P + Q \cos(\theta)}

Time Derivative of a Unit Vector with Constant Magnitude

Conditions: A unit vector n^(t) rotating in a plane with angular velocity $\omega = d\theta/dt$.

Express the unit vector in terms of polar coordinates in a plane: $\hat{n}(t) = \cos(\theta) \hat{i} + \sin(\theta) \hat{j}$.
Differentiate with respect to time t using the chain rule: d$n^/dt = -\sin(\theta) (d\theta/dt) \hat{i} + \cos(\theta) (d\theta/dt) \hat{j}$.
Substitute dθ/d$t = \omega$ and factor out ω: d$n^/dt = \omega (-\sin(\theta) \hat{i} + \cos(\theta) \hat{j})$.
Observe that the vector (-sin(θ)i^ + cos(θ)j^) is perpendicular to n^(t), pointing in the direction of increasing θ.
Represent this geometrically as d$n^/dt = \omega \times \hat{n}$, where ω is the angular velocity vector perpendicular to the plane of rotation.

Result: \frac{d\hat{n}}{dt} = \vec{\omega} \times \hat{n}

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Scalar Dot Product	$$\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z = AB \cos(\theta)$$	A, B = Vector magnitudes, $\theta = Angle between vectors$ Product of individual units	Valid in 3D Euclidean space
Vector Cross Product	$$\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$$	A_x, A_y, A_z and B_x, B_y, $B_z = Cartesian components$ Product of individual units	Right-handed Cartesian coordinate system
Position Vector	$$\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$$	x, y, z = Spatial coordinates Meters (m)	Cartesian reference frame
Direction Cosines	$$l = \cos(\alpha) = \frac{A_x}{A}, \quad m = \cos(\beta) = \frac{A_y}{A}, \quad n = \cos(\gamma) = \frac{A_z}{A}$$	l, m, $n = Direction cosines; \alpha$, β, $\gamma = Angles with axes$ Dimensionless	Non-zero vector A; l^2 + m^2 + $n^2 = 1$
Angular Momentum Vector	$$\vec{L} = \vec{r} \times \vec{p} = m (\vec{r} \times \vec{v})$$	r = Position vector, p = Linear momentum, m = Mass, v = Velocity kg m² s&sup{-1}	Evaluated about a specific origin point
Central Force Vector	$$\vec{F}(r) = f(r) \hat{r} = f(r) \frac{\vec{r}}{r}$$	f(r) = Magnitude function of distance r, r^ = Radial unit vector Newtons (N)	Force acts along the line joining the center and the particle
Coriolis Force Vector	$$\vec{F}_{cor} = -2m (\vec{\omega} \times \vec{v}_r)$$	m = Mass, $\omega = Angular velocity of rotating frame$, $v_r = Relative velocity$ Newtons (N)	Non-inertial rotating reference frame
Gravitational Force Vector	$$\vec{F}_{12} = -\frac{G m_1 m_2}{r^3} \vec{r}_{12}$$	G = Universal gravitational constant, $r_12 = Displacement vector from mass 1$ to mass 2 Newtons (N)	Point masses separated by distance r

Graphs and experiments

Parallelogram Law of Vector Addition Graphical Setup

Draw: Two vectors A and B originating from a common point, completed into a parallelogram with the diagonal representing the resultant vector $R = A + B$.

Interpret: Demonstrates that vector addition is geometric composition independent of coordinate choice, showing the dependence of the resultant magnitude on the included angle θ.

Exam use: Tested conceptually in finding maximum and minimum resultants when vectors act in the same or opposite directions.

Resolution of a Vector into Rectangular Components

Draw: A vector r in the X-Y plane with projections along the X and Y axes forming a right triangle with angle θ relative to the X-axis.

Interpret: Illustrates orthogonal decomposition where $r_x = r \cos(\theta)$ and $r_y = r \sin(\theta)$, enabling algebraic manipulation of multidimensional vectors.

Exam use: Directly applied in resolving inclined plane forces, projectile motion velocity components, and equilibrium problems.

Cross Product Right-Hand Rule Visualization

Draw: Vectors A and B drawn from a common origin spanning a plane, with the cross product vector C perpendicular to the plane pointing upward according to right-hand curl.

Interpret: Shows the non-coplanar axial nature of vector products, essential for defining torque, angular momentum, and magnetic forces.

Exam use: Core conceptual questions in TGT/PGT regarding the direction of torque and angular momentum vectors.

Solved numericals

TGT core

If two vectors A and B have magnitudes 6 units and 8 units respectively, and the angle between them is 60°, find the magnitude of their resultant vector.

Identify given values: A = 6, B = 8, $\theta = 60°$.
Recall the formula for the magnitude of the resultant: $R = \sqrt{A^2 + B^2 + 2AB \cos(\theta)}$.
Substitute the values: $R = \sqrt{6^2 + 8^2 + 2(6)(8) \cos(60^\circ)}$.
Evaluate terms: cos(60°) = 0.5, so $R = \sqrt{36 + 64 + (96)(0.5)} = \sqrt{36 + 64 + 48}$.
Sum the terms: R = \sqrt{148} = 2\sqrt{37} \approx 12.17 units.

Answer: 12.17 units

TGT core

Determine the angle between two vectors $A = 3\hat{i} + 4\hat{j} + 5\hat{k}$ and $B = 3\hat{i} + 4\hat{j} - 5\hat{k}$.

Compute the dot product A · $B = (3)(3) + (4)(4) + (5)(-5) = 9 + 16 - 25 = 0$.
Compute the magnitudes: $A = \sqrt{3^2 + 4^2 + 5^2} = \sqrt{50}$ and $B = \sqrt{3^2 + 4^2 + (-5)^2} = \sqrt{50}$.
Use the dot product definition: A \cdot B = AB \cos(\theta) \implies 0 = \sqrt{50} \sqrt{50} \cos(\theta).
Solve for θ: cos(θ) = 0, which implies $\theta = 90°$.

Answer: 90 degrees (orthogonal)

PGT extension

A particle moves with position vector r(t) = a \cos(\omega t) i^ + a \sin(\omega t) j^. Find its velocity vector and show that velocity is perpendicular to the position vector at any instant t.

Differentiate the position vector with respect to time t: $v = dr/dt = d/dt[a \cos(\omega t)] \hat{i} + d/dt[a \sin(\omega t)] \hat{j}$.
Evaluate derivatives: $v = -a\omega \sin(\omega t) \hat{i} + a\omega \cos(\omega t) \hat{j}$.
Calculate the dot product of position vector r and velocity vector v: r \cdot $v = [a \cos(\omega t)][-a\omega \sin(\omega t)] + [a \sin(\omega t)][a\omega \cos(\omega t)]$.
Simplify the expression: r \cdot $v = -a^2 \omega \sin(\omega t)\cos(\omega t) + a^2 \omega \sin(\omega t)\cos(\omega t) = 0$.
Since the dot product is zero, the vectors r and v are perpendicular for all time t.

Answer: $v = -a\omega \sin(\omega t) \hat{i} + a\omega \cos(\omega t) \hat{j}; r \cdot v = 0 (perpendicular)$

Assumptions and limitations

Assumes a flat, three-dimensional Euclidean space where standard linear vector algebra and Cartesian coordinate systems apply.
Limitations arise in general relativity or non-Euclidean spaces where parallel transport of vectors depends on spacetime curvature.

Common misconceptions

Common error: Any quantity with magnitude and direction is strictly a vector.

Correct: A vector must also obey specific coordinate transformation laws (rotation laws), which excludes quantities like finite rotations that do not commute.

Finite rotations are not vectors because their addition depends on the sequence of operations, whereas infinitesimal rotations do behave as vectors.

Common error: The cross product of two vectors is a scalar.

Correct: The cross product of two polar vectors yields an axial vector (pseudo-vector) perpendicular to their plane.

The vector product incorporates direction determined by the right-hand rule, distinguishing it from the scalar dot product.

Quick revision

Resultant of two vectors: $R = \sqrt{A^2 + B^2 + 2AB \cos(\theta)}$.
Dot product gives scalar: A \cdot B = AB \cos(\theta); zero for orthogonal vectors.
Cross product gives axial vector: A \times B = AB \sin(\theta) \hat{n}; zero for parallel vectors.
Axial vectors like torque and angular momentum do not invert under spatial inversion.

Must remember

Physical quantities are classified into scalars (rank zero), vectors (rank one), and tensors of higher ranks.
Polar vectors reverse sign under spatial inversion ($\mathbf{r} arrow -\mathbf{r}$), whereas axial vectors retain their sign.
The scalar dot product yields a scalar ($A B \cos(\theta)$), while the vector cross product yields an axial vector perpendicular to both operands.
Direction cosines satisfy the identity $l^2 + m^2 + n^2 = 1$ where $l = \cos(\alpha)$, $m = \cos(\beta)$, and $n = \cos(\gamma)$.
Angular momentum ($\mathbf{L} = \mathbf{r} \times \mathbf{p}$) and torque ($\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}$) are classic examples of axial vectors.

Glossary

Polar VectorA vector representing linear translation that reverses direction under spatial inversion (parity transformation).

Axial Vector (Pseudo-vector)A vector representing rotational characteristics associated with an axis that does not invert its direction under spatial inversion.

Direction CosinesCosines of the angles made by a vector with the respective Cartesian coordinate axes, denoted by $l$, $m$, and $n$.

Coriolis ForceAn apparent inertial force given by $mathbf{F}_{cor} = -2m (\boldsymbol{omega} \times mathbf{v}_r)$ experienced in a rotating reference frame.

Polar Vectors vs. Axial Vectors

Parameter	Polar Vectors	Axial Vectors
Origin	Linear translation and displacement	Rotation about an axis
Parity Inversion ($\mathbf{r} arrow -\mathbf{r}$)	Reverses sign (changes direction)	Retains sign (does not change direction)
Examples	Displacement, velocity, force, linear momentum	Angular momentum, torque, magnetic field, Coriolis force

Scalar Product vs. Vector Product

Operation	Scalar (Dot) Product	Vector (Cross) Product
Formula	$\mathbf{A} \cdot \mathbf{B} = AB \cos(\theta)$	$\mathbf{A} \times \mathbf{B} = AB \sin(\theta) \hat{n}$
Resultant Type	Scalar quantity	Axial vector perpendicular to $\mathbf{A}$ and $\mathbf{B}$
Commutativity	Commutative ($\mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A}$)	Anti-commutative ($\mathbf{A} \times \mathbf{B} = -(\mathbf{B} \times \mathbf{A})$)

Practice Quiz

Choose an option to reveal the explanation.

1. Which of the following physical quantities is classified as a scalar?

Explanation: Work is defined as the dot product of force and displacement, yielding a scalar quantity, whereas displacement, momentum, and acceleration are vectors.

2. What is the order of tensor rank for a vector quantity?

Explanation: Scalars are tensors of rank zero, vectors are tensors of rank one, and higher-order quantities like stress and strain are tensors of rank two or higher.

3. How do axial vectors behave under spatial inversion ($\mathbf{r} arrow -\mathbf{r}$)?

Explanation: Axial vectors (pseudo-vectors) are defined via cross products of two polar vectors and do not invert their direction under spatial inversion, retaining their sign.

4. Which of the following is an example of an axial vector?

Explanation: Angular momentum ($\mathbf{L} = \mathbf{r} \times \mathbf{p}$) is an axial vector associated with a rotation axis, while velocity, force, and displacement are polar vectors.

5. If $\mathbf{A} \cdot \mathbf{B} = AB$, what is the angle $\theta$ between the two vectors?

Explanation: Since $\mathbf{A} \cdot \mathbf{B} = AB \cos(\theta)$, if it equals $AB$, then $\cos(\theta) = 1$, which implies $\theta = 0^\circ$.

6. What is the magnitude of the cross product $\mathbf{A} \times \mathbf{B}$ when the two vectors are parallel?

Explanation: The cross product magnitude is $AB \sin(\theta)$. For parallel vectors, $\theta = 0^\circ$, so $\sin(0^\circ) = 0$.

7. What is the sum of the squares of the direction cosines ($l^2 + m^2 + n^2$) of any vector?

Explanation: Direction cosines satisfy the fundamental identity $l^2 + m^2 + n^2 = \cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma) = 1$.

8. Which force formula represents the Coriolis force in a rotating reference frame?

Explanation: The Coriolis force in a rotating non-inertial frame is given by $\mathbf{F}_{cor} = -2m (\boldsymbol{\omega} \times \mathbf{v}_r)$.

9. Why are finite rotations not classified as true vectors?

Explanation: Finite rotations depend on the sequence in which they are applied and do not obey the commutative law of vector addition, whereas infinitesimal rotations do.

10. What is the time derivative of a position vector $\mathbf{r}(t)$ expressed in Cartesian coordinates?

Explanation: The time derivative of position vector $\mathbf{r}(t) = x\hat{i} + y\hat{j} + z\hat{k}$ yields $\frac{d\mathbf{r}}{dt} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$, which is the velocity vector.

11. What is the vector product $\hat{i} \times \hat{j}$?

Explanation: Using a right-handed Cartesian coordinate system, the cross product of unit vectors $\hat{i} \times \hat{j} = \hat{k}$.

12. Which of the following represents a central force vector?

Explanation: A central force is directed along the line joining the center and the particle, mathematically expressed as $\mathbf{F}(r) = f(r)\hat{r}$.

13. What does the product rule derivative $\frac{d}{dt}(\mathbf{A} \cdot \mathbf{B})$ equal?

Explanation: The derivative of the dot product follows the standard product rule: $\frac{d}{dt}(\mathbf{A} \cdot \mathbf{B}) = \frac{d\mathbf{A}}{dt} \cdot \mathbf{B} + \mathbf{A} \cdot \frac{d\mathbf{B}}{dt}$.

14. What are the units of angular momentum vector $\mathbf{L}$?

Explanation: Angular momentum is $\mathbf{L} = \mathbf{r} \times \mathbf{p}$, with units of distance $\times$ momentum = $\text{m} \times (\text{kg} \, \text{m} \, \text{s}^{-1}) = \text{kg} \, \text{m}^2 \, \text{s}^{-1}$.

15. If two vectors $\mathbf{A}$ and $\mathbf{B}$ are perpendicular to each other, what is their dot product?

Explanation: If vectors are perpendicular, $\theta = 90^\circ$, so $\mathbf{A} \cdot \mathbf{B} = AB \cos(90^\circ) = 0$.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. If a vector $\mathbf{A} = 2\hat{i} + 3\hat{j} + \hat{k}$ and $\mathbf{B} = \hat{i} - \hat{j} + 2\hat{k}$, what is their scalar dot product $\mathbf{A} \cdot \mathbf{B}$?

Explanation: Component-wise multiplication gives $\mathbf{A} \cdot \mathbf{B} = (2)(1) + (3)(-1) + (1)(2) = 2 - 3 + 2 = 1$.

2. Which of the following transformation properties distinguishes a polar vector from an axial vector?

Explanation: Polar vectors reverse their sign under spatial inversion ($\mathbf{r} arrow -\mathbf{r}$), whereas axial vectors maintain their sign under spatial inversion.

3. What is the torque vector $\boldsymbol{\tau}$ acting on a particle given position vector $\mathbf{r} = 2\hat{i} \text{ m}$ and force vector $\mathbf{F} = 10\hat{j} \text{ N}$?

Explanation: Torque is $\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F} = (2\hat{i}) \times (10\hat{j}) = 20(\hat{i} \times \hat{j}) = 20\hat{k} \text{ N m}$.

4. If the direction cosines of a vector are $l = 1/2$ and $m = 1/2$, what is the possible value of the third direction cosine $n$ in a 3D Cartesian coordinate system?

Explanation: Using $l^2 + m^2 + n^2 = 1$, we get $(1/2)^2 + (1/2)^2 + n^2 = 1 \implies 1/4 + 1/4 + n^2 = 1 \implies n^2 = 1/2 \implies n = \sqrt{2}/2$ or similar. Wait, let's re-verify: $1/4 + 1/4 = 1/2$, so $n^2 = 1 - 1/2 = 1/2$, hence $n = 1/\sqrt{2}$ or $\sqrt{2}/2$. Wait, looking at options: $\sqrt{3}/2$ is incorrect; let's check if the option should be $1/\sqrt{2}$. Since $1/\sqrt{2} = \sqrt{2}/2$, let's ensure option B is chosen if it's $\sqrt{2}/2$.

5. What is the angle between two vectors $\mathbf{A}$ and $\mathbf{B}$ if the magnitude of their cross product is equal to the dot product ($mid\mathbf{A} \times \mathbf{B}\mid = \mathbf{A} \cdot \mathbf{B}$)?

Explanation: $AB \sin(\theta) = AB \cos(\theta) \implies \tan(\theta) = 1 \implies \theta = 45^\circ$.

6. Which mathematical condition must be satisfied for a vector function $\mathbf{r}(t)$ to have a constant magnitude over time?

Explanation: If magnitude $r$ is constant, then $r^2 = \mathbf{r} \cdot \mathbf{r} = \text{constant}$. Differentiating with respect to $t$ gives $2\mathbf{r} \cdot \frac{d\mathbf{r}}{dt} = 0 \implies \mathbf{r} \cdot \frac{d\mathbf{r}}{dt} = 0$.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. Which of the following physical quantities is an example of a tensor of rank higher than one?

Explanation: Stress, strain, and moments of inertia are described by tensors of rank two or higher, whereas velocity, force, and displacement are rank-one vectors.

2. What is the cross product $\mathbf{A} \times \mathbf{B}$ of two vectors if $\mathbf{A} = 2\hat{i}$ and $\mathbf{B} = 3\hat{j}$?

Explanation: Evaluating the determinant or unit vector cross product: $(2\hat{i}) \times (3\hat{j}) = 6(\hat{i} \times \hat{j}) = 6\hat{k}$.

3. Why does the scalar product of two orthogonal vectors equal zero?

Explanation: The dot product formula is $AB \cos(\theta)$. When vectors are orthogonal, $\theta = 90^\circ$, and $\cos(90^\circ) = 0$, making the product zero.

4. What is the time derivative of vector $\mathbf{A}(t) = t^2\hat{i} + 3t\hat{j} + 5\hat{k}$ at $t = 2 \text{ s}$?

Explanation: Differentiating component-wise: $\frac{d\mathbf{A}}{dt} = 2t\hat{i} + 3\hat{j}$. At $t = 2$, this evaluates to $4\hat{i} + 3\hat{j}$.

5. Which of the following statements about vector addition is correct?

Explanation: Vector addition obeys both commutative ($\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}$) and associative laws.

6. What does the gravitational force vector equation $\mathbf{F}_{12} = -\frac{G m_1 m_2}{r^3} \mathbf{r}_{12}$ signify?

Explanation: The negative sign indicates an attractive force directed along the displacement vector $\mathbf{r}_{12}$, representing Newton's law of gravitation in vector form.

7. If $\mathbf{A} = \hat{i} + \hat{j}$ and $\mathbf{B} = \hat{j} + \hat{k}$, what is the scalar dot product $\mathbf{A} \cdot \mathbf{B}$?

Explanation: $(1)(0) + (1)(1) + (0)(1) = 0 + 1 + 0 = 1$.

8. Which transformation rule defines a true vector quantity?

Explanation: Vectors require specific spatial orientation and must transform under coordinate rotation according to linear transformation rules.

9. What is the physical significance of the vector cross product in mechanics?

Explanation: The cross product generates an axial vector perpendicular to the operand plane, essential for describing rotational phenomena like torque and angular momentum.

10. Under a parity transformation (spatial inversion $\mathbf{r} arrow -\mathbf{r}$), what happens to linear velocity $\mathbf{v}$?

Explanation: Linear velocity is a polar vector representing linear translation, so it reverses sign under spatial inversion ($\mathbf{r} arrow -\mathbf{r}$).