Vectors, magnitude, and components

A vector is a quantity with both magnitude and direction. Examples: displacement of a particle, velocity, acceleration, force. By contrast, a scalar has magnitude only , mass, temperature, energy.

Two vectors are equal if they have the same magnitude and the same direction, regardless of where in space they are drawn. So a vector is genuinely a "free arrow" , translating it does not change it.

Geometric vs algebraic

Geometric: a vector $\vec{a}$ is an arrow from a tail point to a head point. Its length is the magnitude $|\vec{a}|$; the direction is the orientation of the arrow.

Algebraic (in $\mathbb{R}^3$): a vector is a triple $\vec{a} = (a_1, a_2, a_3)$, or equivalently $a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}$, where $\hat{i}, \hat{j}, \hat{k}$ are unit vectors along the coordinate axes.

The two views are equivalent. Given an arrow from $P$ to $Q$, the components are $(Q_1 - P_1, Q_2 - P_2, Q_3 - P_3)$.

Position vectors

The position vector of a point $P = (x, y, z)$ with respect to the origin $O$ is $\vec{OP} = x\hat{i} + y\hat{j} + z\hat{k}$. We typically write $\vec{r}$ for the position vector of a generic point.

If two points $A$ and $B$ have position vectors $\vec{a}$ and $\vec{b}$, then $\vec{AB} = \vec{b} - \vec{a}$ (head minus tail).

Magnitude

For $\vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}$: $|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}.$ This is the Euclidean length. A unit vector has magnitude $1$. Given any non-zero $\vec{a}$, the unit vector in its direction is $\hat{a} = \vec{a}/|\vec{a}|$.

Direction cosines and direction ratios

The direction cosines of $\vec{a}$ are the cosines of the angles $\vec{a}$ makes with the positive $x$, $y$, $z$ axes: $l = \cos\alpha = \frac{a_1}{|\vec{a}|}, \quad m = \cos\beta = \frac{a_2}{|\vec{a}|}, \quad n = \cos\gamma = \frac{a_3}{|\vec{a}|}.$ They satisfy the identity $l^2 + m^2 + n^2 = 1.$

Any triple proportional to $(l, m, n)$ , e.g. the components $a_1, a_2, a_3$ themselves , gives the direction ratios. Direction ratios are not unique (scaling preserves them), but direction cosines are unique up to the sign of $\vec{a}$.

Types of vectors

• Zero vector $\vec{0}$: magnitude $0$, direction undefined.
• Unit vector: magnitude $1$.
• Equal vectors: same magnitude and direction.
• Opposite (negative) vectors: same magnitude, opposite direction; $\vec{a}$ and $-\vec{a}$.
• Parallel/collinear vectors: same or opposite direction. Algebraically, $\vec{a} \parallel \vec{b}$ iff $\vec{b} = \lambda\vec{a}$ for some scalar $\lambda$.

Worked examples

Example 1. Find the magnitude of $\vec{a} = 3\hat{i} - 4\hat{j} + 12\hat{k}$.

$|\vec{a}| = \sqrt{9 + 16 + 144} = \sqrt{169} = 13$.

Example 2. Find a unit vector in the direction of $\vec{a} = 2\hat{i} + 3\hat{j} + 6\hat{k}$.

$|\vec{a}| = \sqrt{4 + 9 + 36} = 7$. So $\hat{a} = \dfrac{2\hat{i} + 3\hat{j} + 6\hat{k}}{7}$.

Example 3. Find the direction cosines of $\vec{a} = \hat{i} - 2\hat{j} + 2\hat{k}$.

$|\vec{a}| = \sqrt{1 + 4 + 4} = 3$. Direction cosines: $(1/3, -2/3, 2/3)$. Check: $1/9 + 4/9 + 4/9 = 9/9 = 1$.

Example 4. If $A = (1, 2, 3)$ and $B = (4, 5, 6)$, find $\vec{AB}$ and $|\vec{AB}|$.

$\vec{AB} = B - A = (3, 3, 3) = 3(\hat{i} + \hat{j} + \hat{k})$. Magnitude: $3\sqrt{3}$.

Example 5. Are $\vec{a} = 2\hat{i} - 4\hat{j} + 6\hat{k}$ and $\vec{b} = -\hat{i} + 2\hat{j} - 3\hat{k}$ parallel?

$\vec{a} = -2\vec{b}$, so yes , anti-parallel (opposite direction).

Example 6. A vector $\vec{r}$ has magnitude $\sqrt{14}$ and direction cosines proportional to $1 : 2 : 3$. Find $\vec{r}$.

Direction ratios $(1, 2, 3)$; $|(1,2,3)| = \sqrt{14}$. So direction cosines: $(1/\sqrt{14}, 2/\sqrt{14}, 3/\sqrt{14})$. Vector $\vec{r} = \sqrt{14} \cdot (l, m, n) = (1, 2, 3) = \hat{i} + 2\hat{j} + 3\hat{k}$.

Try it yourself

1. Find the magnitude of $\vec{a} = 5\hat{i} - 12\hat{j}$.
2. Find a unit vector in the direction of $4\hat{i} + 3\hat{j}$.
3. Direction cosines of $\hat{i} + \hat{j} + \hat{k}$.
4. If $A = (2, 1, 0)$, $B = (-1, 3, 4)$, find $\vec{AB}$ and $|\vec{AB}|$.
5. Are $(2, -4, 6)$ and $(1, -2, 3)$ collinear vectors?
6. Find a vector of magnitude $5$ parallel to $\hat{i} + 2\hat{j} + 2\hat{k}$.
7. Verify that the direction cosines of $\hat{i}$ are $(1, 0, 0)$.
8. Find $\vec{a}$ if $|\vec{a}| = 7$ and direction ratios are $2 : 3 : 6$.
9. Show that $(2, 3, -1)$ and $(-4, -6, 2)$ are anti-parallel.
10. Find the angle which $\hat{i} + \hat{j}$ makes with the $x$-axis.
11. Find a unit vector parallel to $\vec{a} - \vec{b}$ if $\vec{a} = \hat{i} + 2\hat{j} - \hat{k}$, $\vec{b} = 2\hat{i} - \hat{j} + 3\hat{k}$.
12. Find direction cosines of a line equally inclined to all three axes.
13. Find $\vec{r}$ with $|\vec{r}| = 9$ and direction $\hat{i} - 2\hat{j} + 2\hat{k}$.
14. A particle is at $(1, 2, 3)$ and moves to $(4, -2, 8)$. Find the displacement vector and its magnitude.

Pitfalls and tricks

• Direction cosines square-sum to 1; direction ratios do not. Don't confuse them.
• A vector is independent of its starting point. It's a free arrow, not a tethered one.
• Zero vector has undefined direction. Avoid dividing by $|\vec{0}|$.
• Equally-inclined direction: each direction cosine is $\pm 1/\sqrt{3}$.
• Parallel vs equal: parallel vectors share direction but may have any magnitude; equal vectors have both same magnitude and direction.