Vector Algebra Formula Sheet - NDA Exam Study Material

Mastering Vector Algebra is essential for clearing the National Defence Academy (NDA) written examination conducted by the UPSC. Vector Algebra carries significant weightage in the Mathematics paper, consistently contributing 8 to 10 questions (20 to 25 marks). This comprehensive LibreTexts-style study guide covers essential concepts, formulas, geometric interpretations, and shortcut strategies tailored for NDA aspirants.

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1. Basics and Vector Components

A vector in three-dimensional space represents both magnitude and direction. Let a position vector \(\vec{a}\) be defined in component form as \(\vec{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}\) or \(\vec{a} = \langle a_1, a_2, a_3 \rangle\).

Magnitude (Norm)

The length or magnitude of vector \(\vec{a}\) is given by:

\[\|\vec{a}\| = \sqrt{a_1^2 + a_2^2 + a_3^2}\]

Unit Vector

A vector with a magnitude of unity in the direction of \(\vec{a}\) is denoted by \(\hat{a}\):

\[\hat{a} = \frac{\vec{a}}{\|\vec{a}\|} \quad \text{where } \|\vec{a}\| eq 0\]

Position Vector Between Two Points

For points \(P(x_1, y_1, z_1)\) and \(Q(x_2, y_2, z_2)\), the directed segment vector \(\vec{PQ}\) is:

\[\vec{PQ} = (x_2 - x_1)\mathbf{i} + (y_2 - y_1)\mathbf{j} + (z_2 - z_1)\mathbf{k} = \langle x_2-x_1, y_2-y_1, z_2-z_1 \rangle\]

Direction Cosines and Direction Ratios

If \(\alpha, \beta, \gamma\) are the angles that vector \(\vec{a}\) makes with the positive x, y, and z axes respectively, the direction cosines \(l, m, n\) are defined as:

\[l = \cos\alpha = \frac{a_1}{\|\vec{a}\|}, \quad m = \cos\beta = \frac{a_2}{\|\vec{a}\|}, \quad n = \cos\gamma = \frac{a_3}{\|\vec{a}\|}\]

Key identity for direction cosines:

\[\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1 \quad \text{or} \quad l^2 + m^2 + n^2 = 1\]

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2. Dot Product (Scalar Product)

The dot product evaluates the scalar projection of one vector onto another. Algebraically and geometrically, it is defined as:

\[\vec{a} \cdot \vec{b} = \|\vec{a}\| \|\vec{b}\| \cos\theta\]

where \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\) (\(0 \le \theta \le \pi\)).

Component Form

\[\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3\]

Condition for Orthogonality

Two non-zero vectors \(\vec{a}\) and \(\vec{b}\) are perpendicular (orthogonal) if and only if their scalar product equals zero:

\[\vec{a} \perp \vec{b} \iff \vec{a} \cdot \vec{b} = 0\]

Angle Between Two Vectors

\[\cos\theta = \frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\| \|\vec{b}\|} = \frac{a_1 b_1 + a_2 b_2 + a_3 b_3}{\sqrt{a_1^2 + a_2^2 + a_3^2} \sqrt{b_1^2 + b_2^2 + b_3^2}}\]

Scalar and Vector Projections

• Scalar Projection of \(\vec{a}\) on \(\vec{b}\): \(\text{comp}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{\|\vec{b}\|}\)
• Vector Projection of \(\vec{a}\) on \(\vec{b}\): \(\text{proj}_{\vec{b}}\vec{a} = (\frac{\vec{a} \cdot \vec{b}}{\|\vec{b}\|^2})\vec{b}\)

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3. Cross Product (Vector Product)

The cross product yields a vector perpendicular to both input vectors, forming a right-handed system:

\[\vec{a} \times \vec{b} = (\|\vec{a}\| \|\vec{b}\| \sin\theta)\hat{n}\]

where \(\hat{n}\) is the unit normal vector determined by the right-hand rule.

Determinant Form

\[\vec{a} \times \vec{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}\]

Parallelism Condition

Two non-zero vectors are parallel or collinear if and only if their vector product is the zero vector:

\[\vec{a} \parallel \vec{b} \iff \vec{a} \times \vec{b} = \vec{0}\]

Geometric Applications

• Area of Parallelogram: Formed by adjacent vector sides \(\vec{a}\) and \(\vec{b}\): \(\text{Area} = \|\vec{a} \times \vec{b}\|\)
• Area of Triangle: Formed by vectors \(\vec{a}\) and \(\vec{b}\): \(\text{Area} = \frac{1}{2}\|\vec{a} \times \vec{b}\|\)

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4. Triple Products

Scalar Triple Product (Box Product)

The scalar triple product represents the dot product of one vector with the cross product of the other two:

\[[\vec{a} \ \vec{b} \ \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}\]

• Geometric Meaning: Represents the volume of a parallelepiped with coterminous edges \(\vec{a}, \vec{b}, \vec{c}\).
• Coplanarity Condition: Vectors \(\vec{a}, \vec{b}, \vec{c}\) are coplanar if and only if \([\vec{a} \ \vec{b} \ \vec{c}] = 0\).

Vector Triple Product

The vector triple product expands using the standard expansion identity (BAC-CAB rule):

\[\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}\]

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5. Overview of Vector Calculus

Vector Differentiation

For a position vector function \(\vec{r}(t) = x(t)\mathbf{i} + y(t)\mathbf{j} + z(t)\mathbf{k}\):

\[\frac{d}{dt}\vec{r}(t) = \langle x'(t), y'(t), z'(t) \rangle\]

Gradient, Divergence, and Curl Operators

• Gradient of Scalar Function \(f\): \( abla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k}\)
• Divergence of Vector Field \(\vec{F}\): \( abla \cdot \vec{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}\)
• Curl of Vector Field \(\vec{F}\): \[ abla \times \vec{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_1 & F_2 & F_3 \end{vmatrix}\]

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6. NDA PYQ Analysis & Weightage

Vector Algebra is one of the most scoring chapters in the UPSC NDA Mathematics Paper. Here is an analysis based on recent NDA exam patterns:

• Expected Number of Questions: 8 to 10 Questions
• Total Marks Contribution: 20 to 25 Marks
• High-Frequency Topics: Scalar Projection, Coplanarity via Scalar Triple Product, Section Formula applications, and Work Done/Torque problems.
• UPSC Pattern Insight: Statement-based questions testing conditions for orthogonality (\(\vec{a} \cdot \vec{b} = 0\)) and collinearity (\(\vec{a} \times \vec{b} = \vec{0}\)) are heavily emphasized.

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7. Frequently Asked Questions (People Also Ask)