Vector Algebra Class 12: notes and important questions

Vector Algebra — revision notes

1. Basics

• \(\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}\): magnitude \(|\vec{r}| = \sqrt{x^2 + y^2 + z^2}\), unit vector \(\hat{r} = \dfrac{\vec{r}}{|\vec{r}|}\).
• \(\overrightarrow{AB} = \vec{b} - \vec{a}\) (position vector of head minus tail).
• Direction cosines \(l = \dfrac{x}{r}, m = \dfrac{y}{r}, n = \dfrac{z}{r}\) with \(l^2 + m^2 + n^2 = 1\); direction ratios are any multiple \((x, y, z)\).
• Collinear (parallel) vectors: \(\vec{b} = \lambda\vec{a}\), i.e. components in the same ratio.

2. Section formula

Point dividing \(AB\) internally in \(m : n\): \(\dfrac{n\vec{a} + m\vec{b}}{m + n}\); externally: \(\dfrac{m\vec{b} - n\vec{a}}{m - n}\); midpoint \(\dfrac{\vec{a} + \vec{b}}{2}\).

3. Scalar (dot) product

• \(\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3\); \(\theta \in [0, \pi]\).
• \(\vec{a} \perp \vec{b} \iff \vec{a}\cdot\vec{b} = 0\); \(\vec{a}\cdot\vec{a} = |\vec{a}|^2\).
• Projection of \(\vec{a}\) on \(\vec{b}\): \(\dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|}\); projection vector \(\left(\dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|^2}\right)\vec{b}\).
• \(|\vec{a} \pm \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 \pm 2\vec{a}\cdot\vec{b}\) — the key tool for magnitude problems.

4. Vector (cross) product

• \(\vec{a}\times\vec{b} = |\vec{a}||\vec{b}|\sin\theta\,\hat{n}\) (right-hand rule) \(= \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}\).
• \(\vec{b}\times\vec{a} = -\vec{a}\times\vec{b}\); \(\vec{a}\times\vec{a} = \vec{0}\); \(\vec{a}\parallel\vec{b} \iff \vec{a}\times\vec{b} = \vec{0}\).
• Area of parallelogram \(= |\vec{a}\times\vec{b}|\) (adjacent sides) \(= \tfrac12|\vec{d}_1\times\vec{d}_2|\) (diagonals); area of triangle \(ABC = \tfrac12|\overrightarrow{AB}\times\overrightarrow{AC}|\).
• Unit vector perpendicular to \(\vec{a}\) and \(\vec{b}\): \(\pm\dfrac{\vec{a}\times\vec{b}}{|\vec{a}\times\vec{b}|}\). Also \(|\vec{a}\times\vec{b}|^2 + (\vec{a}\cdot\vec{b})^2 = |\vec{a}|^2|\vec{b}|^2\).

Worked example 1

\(|\vec{a}| = 2\), \(|\vec{b}| = 3\), \(\vec{a}\cdot\vec{b} = 3\): \(|\vec{a} - \vec{b}|^2 = 4 + 9 - 6 = 7\), so \(|\vec{a} - \vec{b}| = \sqrt7\); \(\cos\theta = \tfrac12\), \(\theta = \tfrac\pi3\).

Worked example 2

Area of the triangle with vertices \((0, 0, 0)\), \((1, 2, 0)\), \((0, 1, 3)\): \((\hat{i} + 2\hat{j})\times(\hat{j} + 3\hat{k}) = 6\hat{i} - 3\hat{j} + \hat{k}\), area \(= \tfrac12\sqrt{46}\).

Worked example 3

Point dividing \(A(2, 1, 3)\), \(B(5, 7, 0)\) in \(1 : 2\): \(\dfrac{2(2, 1, 3) + (5, 7, 0)}{3} = (3, 3, 2)\).

Common errors

• Middle term of the cross-product determinant: it is \(-\hat{j}(a_1b_3 - a_3b_1)\).
• Using \(\tfrac{m\vec{a} + n\vec{b}}{m + n}\) (weights swapped) in the section formula.
• Giving only one sign for a unit perpendicular vector, or the wrong one when a condition (acute/obtuse with an axis) is given.
• Confusing the scalar projection (a number) with the projection vector.
• Writing \(|\vec{a} + \vec{b}| = |\vec{a}| + |\vec{b}|\).

Board-exam tips

• For “find the angle” questions, write \(\cos\theta = \dots\) and the final \(\theta = \cos^{-1}(\cdot)\) if it is not standard.
• Scalar triple product and coplanarity are not in the current syllabus.