Mathematics · Ch 10 — Vector Algebra

Summary

Vector definition: A quantity with both magnitude and direction, denoted as $\vec{a}$ or $\mathbf{a}$ . Its magnitude is $|\vec{a}|$ .
Position vector: $\vec{OP}$ of point $P(x,y,z)$ is $x\hat{i} + y\hat{j} + z\hat{k}$ .
Direction cosines: Cosines of angles with axes: $l = \frac{x}{|\vec{r}|}$ , $m = \frac{y}{|\vec{r}|}$ , $n = \frac{z}{|\vec{r}|}$ , with $l^2 + m^2 + n^2 = 1$ .
Vector addition: Triangle law: $\vec{AB} + \vec{BC} = \vec{AC}$ . Parallelogram law: $\vec{a} + \vec{b}$ is diagonal.
Scalar multiplication: $k\vec{a}$ scales magnitude by $|k|$ ; direction same if $k>0$ , opposite if $k<0$ .
Dot product: $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3$ . Result is a scalar.
Cross product: $\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \, \hat{n}$ , with direction by right-hand rule. In components: $\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$ .
Projection: Scalar projection of $\vec{a}$ on $\vec{b}$ is $\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$ ; vector projection is $\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right)\vec{b}$ .
Area: $|\vec{a} \times \vec{b}|$ gives area of parallelogram; $\frac{1}{2}|\vec{a} \times \vec{b}|$ gives area of triangle. …