Vector Algebra — chapter notes

A scalar has magnitude only (mass, time, temperature, speed). A vector has magnitude and direction (displacement, velocity, force). A vector is drawn as a directed line segment $\overrightarrow{AB}$ from the initial point $A$ to the terminal point $B$; its length $|\overrightarrow{AB}|$ is its magnitude.

Types: zero vector ($|\vec{0}| = 0$, direction arbitrary); unit vector (magnitude $1$); coinitial vectors (same initial point); collinear vectors (parallel to one line, whatever their magnitudes or senses); equal vectors (same magnitude and direction); negative of a vector (same magnitude, opposite direction): $\overrightarrow{BA} = -\overrightarrow{AB}$.

With unit vectors $\hat{i}, \hat{j}, \hat{k}$ along the axes, $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ has components $x, y, z$ and magnitude $|\vec{r}| = \sqrt{x^2 + y^2 + z^2}$. The unit vector along $\vec{a}$ is $\hat{a} = \dfrac{\vec{a}}{|\vec{a}|}$.

If $\vec{r}$ makes angles $\alpha, \beta, \gamma$ with the positive axes, its direction cosines are $l = \cos\alpha = \dfrac{x}{|\vec{r}|}$, $m = \cos\beta = \dfrac{y}{|\vec{r}|}$, $n = \cos\gamma = \dfrac{z}{|\vec{r}|}$, and $l^2 + m^2 + n^2 = 1$. Numbers proportional to $l, m, n$ (such as $x, y, z$) are direction ratios.

Example: $6\hat{i} - 2\hat{j} + 3\hat{k}$ has magnitude $7$ and direction cosines $\dfrac67, -\dfrac27, \dfrac37$.

Triangle law: $\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}$. Parallelogram law: for coinitial $\vec{a}$, $\vec{b}$, the diagonal of the parallelogram on them is $\vec{a} + \vec{b}$. Addition is commutative and associative; $\vec{a} + \vec{0} = \vec{a}$; $\vec{a} + (-\vec{a}) = \vec{0}$. Around a closed triangle $\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CA} = \vec{0}$.

Scalar multiple: $\lambda\vec{a}$ has magnitude $|\lambda||\vec{a}|$, the same direction as $\vec{a}$ if $\lambda > 0$ and the opposite if $\lambda < 0$. In components, add (or multiply) component by component.

Collinear test: $\vec{a}$ and $\vec{b}$ are collinear iff $\vec{b} = \lambda\vec{a}$, i.e. $\dfrac{b_1}{a_1} = \dfrac{b_2}{a_2} = \dfrac{b_3}{a_3}$ (zero components must match).

The position vector of $P(x, y, z)$ is $\overrightarrow{OP} = x\hat{i} + y\hat{j} + z\hat{k}$. For any two points, $\overrightarrow{AB} = \vec{b} - \vec{a}$ ("head minus tail").

If $R$ divides $PQ$ in the ratio $m : n$:

• internally: $\vec{r} = \dfrac{m\vec{q} + n\vec{p}}{m + n}$;
• externally: $\vec{r} = \dfrac{m\vec{q} - n\vec{p}}{m - n}$;
• midpoint: $\vec{r} = \dfrac{\vec{p} + \vec{q}}{2}$.

Collinear points $A, B, C$: show $\overrightarrow{AB} = \lambda\overrightarrow{AC}$ (parallel vectors with the common point $A$).

Example: $P(1, 3, -2)$, $Q(4, -3, 7)$, ratio $1 : 2$: internal point $(2, 1, 1)$, external point $(-2, 9, -11)$.

$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3$, a scalar, with $0 \le \theta \le \pi$.

Properties: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$; $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$; $\vec{a} \cdot \vec{a} = |\vec{a}|^2$; $\hat{i} \cdot \hat{i} = 1$, $\hat{i} \cdot \hat{j} = 0$.

Angle: $\cos\theta = \dfrac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$. Perpendicular (non-zero vectors): $\vec{a} \cdot \vec{b} = 0$. Positive dot product: acute angle; negative: obtuse.

Projection of $\vec{a}$ on $\vec{b}$: $\dfrac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$; the vector component of $\vec{a}$ along $\vec{b}$ is this number times $\hat{b}$.

Expansions: $|\vec{a} \pm \vec{b}|^2 = |\vec{a}|^2 \pm 2\,\vec{a} \cdot \vec{b} + |\vec{b}|^2$; $(\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = |\vec{a}|^2 - |\vec{b}|^2$.

$\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta\,\hat{n}$, a vector perpendicular to both $\vec{a}$ and $\vec{b}$ (right-hand rule).

Components: $\vec{a} \times \vec{b} = (a_2b_3 - a_3b_2)\hat{i} + (a_3b_1 - a_1b_3)\hat{j} + (a_1b_2 - a_2b_1)\hat{k}$.

Properties: $\vec{b} \times \vec{a} = -\vec{a} \times \vec{b}$; $\vec{a} \times \vec{a} = \vec{0}$; non-zero $\vec{a} \times \vec{b} = \vec{0}$ iff $\vec{a} \parallel \vec{b}$; $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, $\hat{k} \times \hat{i} = \hat{j}$; $|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2|\vec{b}|^2$.

Unit vectors perpendicular to both: $\pm\dfrac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|}$.

Areas: parallelogram with adjacent sides $\vec{a}, \vec{b}$: $|\vec{a} \times \vec{b}|$; with diagonals $\vec{d_1}, \vec{d_2}$: $\dfrac12|\vec{d_1} \times \vec{d_2}|$; triangle $ABC$: $\dfrac12|\overrightarrow{AB} \times \overrightarrow{AC}|$.

• Length or direction of one vector: magnitude, unit vector, direction cosines.
• Points on a line, ratios: position vectors, section formula, collinearity.
• Angles, right angles, shadows, work done ($W = \vec{F} \cdot \vec{d}$): dot product.
• A direction at right angles to two vectors, areas: cross product.

Check your work: a unit vector has magnitude $1$; direction cosines square-sum to $1$; a cross product dots to $0$ with both factors; an area is positive.