Vector Algebra — NDA Mathematics

A vector question at NDA level is rarely conceptually hard — it is a components-in, formula-out exercise. The entire chapter runs on fewer than fifteen formulas: one for magnitude, one for addition, two for the dot product, three for the cross product, and a handful of applications built directly on top. Learn the formula, learn which product (dot or cross) the wording is pointing at, and the arithmetic does the rest.

1. What NDA actually asks

Vector Algebra carries weightPct 6 of NDA Mathematics — roughly 7–8 of the 120 questions, worth about 18–20 marks. It is a self-contained, low-ambiguity topic: nearly every question falls into one of six recurring types.

1. Magnitude, unit vectors and direction cosines — given components, find $|\vec{a}|$, $\hat{a}$, or the direction cosines $(l,m,n)$.
2. Vector addition — triangle/parallelogram law, resultant magnitude and direction for two vectors at a given angle.
3. Position vectors and section formula — midpoint, centroid of a triangle, or a point dividing a segment in a given ratio, expressed through position vectors.
4. Dot product applications — angle between two vectors, testing perpendicularity, scalar/vector projection of one vector onto another.
5. Cross product applications — area of a triangle or parallelogram, a unit vector perpendicular to two given vectors, testing whether two vectors are parallel.
6. Scalar triple product — coplanarity of three vectors, volume of a parallelepiped/tetrahedron (asked less often, but a guaranteed easy mark when it appears, since it is pure determinant evaluation).

The chapter is graphics-light and formula-heavy — there are no diagrams to interpret in the exam, just components to plug into the right identity.

2. Vector basics & operations

Representation. A vector in the plane is $\vec{a}=a_1\hat{i}+a_2\hat{j}$; in space, $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$. Its magnitude is

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

Unit vector along $\vec{a}$: $\hat{a}=\frac{\vec{a}}{|\vec{a}|}$. NDA loves handing you a vector and asking for "a unit vector in the direction of $\vec{a}$" — divide every component by the magnitude, nothing more.

Direction cosines. If $\vec{a}$ makes angles $\alpha ,\beta ,\gamma$ with the $x$, $y$, $z$ axes, its direction cosines are $l=\cos \alpha =\frac{a_1}{|\vec{a}|}$, $m=\frac{a_2}{|\vec{a}|}$, $n=\frac{a_3}{|\vec{a}|}$, and always

$l^2+m^2+n^2=1$

Direction ratios are just any set proportional to $(l,m,n)$ — the components themselves qualify; only direction cosines need the magnitude-normalisation.

Position vectors and section formula. For points $A(\vec{a})$ and $B(\vec{b})$, the point dividing $AB$ in ratio $m:n$ has position vector

$\vec{r}=\frac{m\vec{b}+n\vec{a}}{m+n}(\text{internal}),\vec{r}=\frac{m\vec{b}-n\vec{a}}{m-n}(\text{external})$

Midpoint of $AB$: $\frac{\vec{a}+\vec{b}}{2}$. Centroid of a triangle with vertices $\vec{a},\vec{b},\vec{c}$: $\frac{\vec{a}+\vec{b}+\vec{c}}{3}$ — a very frequent one-line NDA question.

Addition — triangle and parallelogram law. $\vec{a}+\vec{b}$ is found by placing vectors head-to-tail (triangle law) or as the diagonal of the parallelogram they span (parallelogram law); both give the same resultant. For two vectors of magnitude $|\vec{a}|,|\vec{b}|$ with angle $\theta$ between them:

$|\vec{a}+\vec{b}|=\sqrt{|\vec{a}{|}^2+|\vec{b}{|}^2+2|\vec{a}||\vec{b}|\cos \theta }$

and the resultant makes angle $\alpha$ with $\vec{a}$ where $\tan \alpha =\frac{|\vec{b}|\sin \theta }{|\vec{a}|+|\vec{b}|\cos \theta }$. At $\theta =90°$ this collapses to plain Pythagoras — NDA sets up 3-4-5 or 5-12-13 triples here constantly.

Special vectors: the zero vector $\vec{0}$ has no defined direction; like vectors point the same way; collinear vectors are scalar multiples of each other ($\vec{a}=k\vec{b}$); coplanar vectors lie in one plane. Scalar multiplication $k\vec{a}$ scales the magnitude by $|k|$ and reverses direction when $k<0$.

3. Dot (scalar) product & applications

$\vec{a}\cdot \vec{b}=|\vec{a}||\vec{b}|\cos \theta =a_1{b}_1+a_2{b}_2+a_3{b}_3$

The output is a scalar, not a vector — this is the single fact NDA distractors exploit most. Properties: commutative ($\vec{a}\cdot \vec{b}=\vec{b}\cdot \vec{a}$), distributive over addition, and $\vec{a}\cdot \vec{a}=|\vec{a}{|}^2$. For the unit vectors: $\hat{i}\cdot \hat{i}=\hat{j}\cdot \hat{j}=\hat{k}\cdot \hat{k}=1$ and $\hat{i}\cdot \hat{j}=\hat{j}\cdot \hat{k}=\hat{k}\cdot \hat{i}=0$.

Angle between two vectors:

$\cos \theta =\frac{\vec{a}\cdot \vec{b}}{|\vec{a}||\vec{b}|}$

Perpendicularity test: $\vec{a}\perp \vec{b}⟺\vec{a}\cdot \vec{b}=0$. This is the fastest identity in the whole chapter — no square roots, no trig tables, just a dot product that must vanish.

Projection. The scalar projection of $\vec{a}$ on $\vec{b}$ is $\frac{\vec{a}\cdot \vec{b}}{|\vec{b}|}$ — a number, which can be negative (angle obtuse). The vector projection is that scalar times the unit vector along $\vec{b}$:

$\text{vector projection of }\vec{a}\text{ on }\vec{b}=\left(\frac{\vec{a}\cdot \vec{b}}{|\vec{b}{|}^2}\right)\vec{b}$

Note the denominator changes from $|\vec{b}|$ to $|\vec{b}{|}^2$ once you multiply by $\vec{b}$ instead of $\hat{b}$ — this is the single most common projection slip.

4. Cross (vector) product & applications

$\vec{a}\times \vec{b}=|\vec{a}||\vec{b}|\sin \theta \hat{n},\hat{n}\perp \vec{a},\vec{b}\text{ (right-hand rule)}$

Unlike the dot product, the output is a vector, and the product is anti-commutative: $\vec{a}\times \vec{b}=-(\vec{b}\times \vec{a})$. In components:

$\vec{a}\times \vec{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right|=(a_2{b}_3-a_3{b}_2)\hat{i}-(a_1{b}_3-a_3{b}_1)\hat{j}+(a_1{b}_2-a_2{b}_1)\hat{k}$

Note the minus sign in front of the $\hat{j}$ term — the single most common arithmetic error in this chapter. For unit vectors, the cyclic order $\hat{i}\to \hat{j}\to \hat{k}\to \hat{i}$ gives $\hat{i}\times \hat{j}=\hat{k}$, $\hat{j}\times \hat{k}=\hat{i}$, $\hat{k}\times \hat{i}=\hat{j}$; reversed order flips the sign, and $\hat{i}\times \hat{i}=\hat{j}\times \hat{j}=\hat{k}\times \hat{k}=\vec{0}$.

Applications:

• Area of the parallelogram with adjacent sides $\vec{a},\vec{b}$: $|\vec{a}\times \vec{b}|$.
• Area of the triangle with sides $\vec{a},\vec{b}$ from a common vertex: $\frac{1}{2}|\vec{a}\times \vec{b}|$. For a triangle with vertices $A,B,C$: $\frac{1}{2}|\overrightarrow{AB}\times \overrightarrow{AC}|$.
• Parallel vectors: $\vec{a}\parallel \vec{b}⟺\vec{a}\times \vec{b}=\vec{0}$.
• Unit vector perpendicular to both $\vec{a}$ and $\vec{b}$: $\hat{n}=\frac{\vec{a}\times \vec{b}}{|\vec{a}\times \vec{b}|}$.

Scalar triple product. $[\vec{a}\vec{b}\vec{c}]=\vec{a}\cdot (\vec{b}\times \vec{c})$, computed directly as the determinant of the three rows of components. It is cyclic — $[\vec{a}\vec{b}\vec{c}]=[\vec{b}\vec{c}\vec{a}]=[\vec{c}\vec{a}\vec{b}]$ — but swapping any two vectors (not a cyclic shift) flips its sign. Geometrically it is the volume of the parallelepiped with edges $\vec{a},\vec{b},\vec{c}$; the volume of the tetrahedron they form is $\frac{1}{6}\left|[\vec{a}\vec{b}\vec{c}]\right|$. Three vectors are coplanar exactly when $[\vec{a}\vec{b}\vec{c}]=0$ — this is the fastest coplanarity test there is.

6. Common traps

• Dot vs cross confusion. Angle/projection questions need $\cos \theta$ and the dot product; area/perpendicular-vector questions need $\sin \theta$ and the cross product. Read the question word — "angle", "projection", "perpendicular test" → dot; "area", "perpendicular vector", "parallel test" → cross.
• Sign slip in the $\hat{j}$ term of the cross-product determinant — it is subtracted, and the 2×2 minor itself needs its own sign handled correctly: $-(a_1{b}_3-a_3{b}_1)$, not $-(a_1{b}_3)-a_3{b}_1$.
• Forgetting anti-commutativity — $\vec{b}\times \vec{a}=-(\vec{a}\times \vec{b})$. If the question asks for $\vec{b}\times \vec{a}$, compute $\vec{a}\times \vec{b}$ and flip every sign; don't just swap the rows mentally and hope.
• Triangle vs parallelogram area — $|\vec{a}\times \vec{b}|$ is the parallelogram's area; the triangle needs an extra $÷2$. This single missing factor of 2 is the most common wrong option on cross-product questions.
• Scalar vs vector projection — scalar projection divides by $|\vec{b}|$ once; vector projection divides by $|\vec{b}{|}^2$ (then multiplies back by $\vec{b}$). Mixing the two gives a vector with the wrong magnitude.
• Treating a zero answer as an error. $\vec{a}\cdot \vec{b}=0$ (perpendicular) and $\vec{a}\times \vec{b}=\vec{0}$ (parallel) and $[\vec{a}\vec{b}\vec{c}]=0$ (coplanar) are all legitimate, exam-favourite outcomes — don't discard them and hunt for a "nicer" number.
• Direction ratios treated as direction cosines — direction ratios need not satisfy $l^2+m^2+n^2=1$; only the normalised (divided-by-magnitude) version does.

7. Revision protocol

This chapter rewards raw formula fluency more than any other topic in NDA Mathematics — there is very little "thinking" once you've identified dot vs cross. Write out the determinant form of the cross product and the dot-product angle formula from memory daily until the sign pattern is automatic. Then drill ten mixed questions where you must first decide which product the wording demands before touching a calculator-free computation — that fifteen-second classification step, not the arithmetic, is what NDA is actually testing.