Vector Algebra — chapter notes

A vector has magnitude and direction and is drawn as a directed line segment $\overrightarrow{PQ}$. For $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$:

• $|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$; the unit vector is $\hat{a} = \dfrac{\vec{a}}{|\vec{a}|}$.
• Direction cosines $l, m, n$ are the cosines of the angles with the axes: $l = \dfrac{a_1}{|\vec{a}|}$ and so on, with $l^2 + m^2 + n^2 = 1$.
• Direction ratios are any three numbers proportional to $l, m, n$, for example $a_1, a_2, a_3$.
• A vector of magnitude $k$ along $\vec{a}$ is $k\hat{a}$.

• Zero vector $\vec{0}$: magnitude $0$, no definite direction.
• Unit vector: magnitude $1$.
• Equal vectors: same magnitude and same direction (wherever drawn).
• Negative of a vector: same magnitude, opposite direction; $\overrightarrow{BA} = -\overrightarrow{AB}$.
• Collinear (parallel) vectors: $\vec{b} = \lambda\vec{a}$ for a scalar $\lambda$ — like when $\lambda > 0$, opposite when $\lambda < 0$. In components, the ratios $\dfrac{b_1}{a_1} = \dfrac{b_2}{a_2} = \dfrac{b_3}{a_3}$ are equal.

The position vector of $A(x, y, z)$ is $\vec{a} = x\hat{i} + y\hat{j} + z\hat{k}$, and $\overrightarrow{AB} = \vec{b} - \vec{a}$ (head minus tail).

• Triangle law: $\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}$; the parallelogram law gives the same sum as a diagonal.
• In components, add, subtract and scale one direction at a time.
• The distance between $A$ and $B$ is $|\overrightarrow{AB}|$.

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

• internally: $\vec{r} = \dfrac{m\vec{q} + n\vec{p}}{m + n}$ ($R$ between $P$ and $Q$);
• externally: $\vec{r} = \dfrac{m\vec{q} - n\vec{p}}{m - n}$, $m \ne n$ ($R$ on $PQ$ produced).
• Midpoint: $\dfrac{\vec{p} + \vec{q}}{2}$. The centroid of a triangle is $\dfrac{\vec{a} + \vec{b} + \vec{c}}{3}$.
• Collinear points: $A, B, C$ are collinear when $\overrightarrow{AC} = \lambda\,\overrightarrow{AB}$; the ratio follows from $\lambda$.

• $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3$, a scalar; $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$; $\vec{a} \cdot \vec{a} = |\vec{a}|^2$.
• Angle: $\cos\theta = \dfrac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$; non-zero vectors are perpendicular iff $\vec{a} \cdot \vec{b} = 0$.
• $|\vec{a} \pm \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 \pm 2\,\vec{a} \cdot \vec{b}$.
• Scalar projection of $\vec{a}$ on $\vec{b}$: $\dfrac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$ (negative for an obtuse angle). Vector projection: $\dfrac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\,\vec{b}$.

• $\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta\,\hat{n}$, perpendicular to both (right-hand rule); $\vec{b} \times \vec{a} = -(\vec{a} \times \vec{b})$.
• $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, $\hat{k} \times \hat{i} = \hat{j}$; $\hat{i} \times \hat{i} = \vec{0}$.
• In components, expand $\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$.
• Unit vectors perpendicular to both: $\pm\dfrac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|}$.
• Areas: parallelogram $|\vec{a} \times \vec{b}|$; triangle $\tfrac12|\vec{a} \times \vec{b}|$; with diagonals $\tfrac12|\vec{d_1} \times \vec{d_2}|$.
• Parallel / collinear: $\vec{a} \times \vec{b} = \vec{0}$. Also $|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2|\vec{b}|^2$.