How a Single Arrow Can Store Both How Far and Which Way

Saying a car travelled 60 km is incomplete — towards Pune or towards Nashik? Displacement, velocity and force all need a direction as well as a size. Vectors package both, and their algebra underlies mechanics, computer graphics and three-dimensional geometry.

This lesson covers vectors as directed line segments, types of vectors and direction cosines, components and operations, and the section formula.

**A vector is represented by a directed line segment $\overrightarrow{AB}$, whose length gives its magnitude $|\overrightarrow{AB}|$ and whose arrow, from the initial point A to the terminal point B, gives its direction.

Key ideas:

- Scalars have size only — mass, time, temperature
- Vectors have size and direction — displacement, velocity, force
- The position vector** of $P(x,y,z)$ is $\overrightarrow{OP}=x\hat{i}+y\hat{j}+z\hat{k}$
- The magnitude of $\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$ is $|\vec{r}|=\sqrt{x^2+y^2+z^2}$

Worked example. For $A(1,2,-1)$ and $B(4,6,11)$:

$$\overrightarrow{AB}=3\hat{i}+4\hat{j}+12\hat{k},|\overrightarrow{AB}|=\sqrt{9+16+144}=13$$

Worked example 2. A drone flies 30 m east and then 40 m north. Its displacement is $30\hat{i}+40\hat{j}$, with magnitude 50 m, at ${\tan }^{-1}\frac{40}{30}\approx 53.1°$ north of east.

An everyday example. Directions to a friend's house as '2 km towards the railway station' describe a vector: a distance together with a direction.

The substance. **$\overrightarrow{AB}$ and $\overrightarrow{BA}$ have the same magnitude but opposite directions** — so $\overrightarrow{BA}=-\overrightarrow{AB}$.

**Vectors are equal when they have the same magnitude and direction, a unit vector has magnitude 1, the zero vector has magnitude 0, and collinear vectors are scalar multiples of each other; the direction cosines of $a\hat{i}+b\hat{j}+c\hat{k}$ are $\frac{a}{r}$, $\frac{b}{r}$ and $\frac{c}{r}$ with $r=\sqrt{a^2+b^2+c^2}$, and a, b, c are its direction ratios.

Types of vectors:

- Equal — same magnitude and direction, wherever they start
- Unit** — magnitude 1; along $\vec{a}$ it is $\hat{a}=\frac{\vec{a}}{|\vec{a}|}$
- Zero — magnitude 0, with no definite direction
- Collinear or parallel — $\vec{b}=\lambda \vec{a}$ for some scalar $\lambda$

Direction cosines. If a vector makes angles $\alpha$, $\beta$ and $\gamma$ with the axes, its direction cosines are $l=\cos \alpha$, $m=\cos \beta$ and $n=\cos \gamma$, with $l^2+m^2+n^2=1$.

Worked example. For $\vec{a}=2\hat{i}-\hat{j}+2\hat{k}$, $|\vec{a}|=3$, so the direction cosines are $\frac{2}{3}$, $-\frac{1}{3}$ and $\frac{2}{3}$, the direction ratios are 2, $-1$ and 2, and the unit vector is $\frac{1}{3}(2\hat{i}-\hat{j}+2\hat{k})$. Check: $\frac{4}{9}+\frac{1}{9}+\frac{4}{9}=1$.

An everyday example. Two cars driving side by side on a straight highway at the same speed have equal velocity vectors.

The substance. Direction ratios are not unique, but direction cosines are — 2, $-1$, 2 and 4, $-2$, 4 describe the same direction.

**Any vector in space can be written as $a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$, and vectors are added, subtracted and multiplied by scalars component by component, which matches the triangle law of addition.

Rules for $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$:**

- $\vec{a}+\vec{b}=(a_1+b_1)\hat{i}+(a_2+b_2)\hat{j}+(a_3+b_3)\hat{k}$
- $\lambda \vec{a}=\lambda a_1\hat{i}+\lambda a_2\hat{j}+\lambda a_3\hat{k}$
- Triangle law — $\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}$

Worked example. Let $\vec{a}=2\hat{i}+3\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}-2\hat{j}+4\hat{k}$.

- $\vec{a}+\vec{b}=3\hat{i}+\hat{j}+3\hat{k}$
- $2\vec{a}-3\vec{b}=\hat{i}+12\hat{j}-14\hat{k}$
- $|\vec{a}+\vec{b}|=\sqrt{9+1+9}=\sqrt{19}\approx 4.36$

Worked example 2 (collinear points). $A(1,2,3)$, $B(2,4,5)$ and $C(4,8,9)$ are collinear, since $\overrightarrow{AB}=\hat{i}+2\hat{j}+2\hat{k}$ and $\overrightarrow{AC}=3\hat{i}+6\hat{j}+6\hat{k}=3\overrightarrow{AB}$.

An everyday example. A boat crossing a river moves with its own velocity plus the current's velocity, and the vector sum sets its actual path.

The substance. **$|\vec{a}+\vec{b}|$ is not $|\vec{a}|+|\vec{b}|$ in general** — they are equal only when the vectors point the same way.

**If A and B have position vectors $\vec{a}$ and $\vec{b}$, the point dividing AB internally in the ratio $m:n$ has position vector $\frac{m\vec{b}+n\vec{a}}{m+n}$, externally $\frac{m\vec{b}-n\vec{a}}{m-n}$, and the midpoint has position vector $\frac{\vec{a}+\vec{b}}{2}$.

Worked example (internal).** For $\vec{a}=\hat{i}+2\hat{j}-\hat{k}$ and $\vec{b}=4\hat{i}-\hat{j}+5\hat{k}$, the point dividing AB in the ratio $2:1$ is

$$\frac{2(4\hat{i}-\hat{j}+5\hat{k})+(\hat{i}+2\hat{j}-\hat{k})}{3}=3\hat{i}+3\hat{k}$$

Worked example (external). Dividing the same segment externally in the ratio $2:1$ gives $\frac{2\vec{b}-\vec{a}}{1}=7\hat{i}-4\hat{j}+11\hat{k}$.

Centroid. The centroid of a triangle with vertices $\vec{a}$, $\vec{b}$ and $\vec{c}$ has position vector $\frac{\vec{a}+\vec{b}+\vec{c}}{3}$.

An everyday example. Placing a support pillar two-thirds of the way along a straight footbridge uses the section formula with the ratio $2:1$.

The substance. The vector section formula is the coordinate formula in compact form — each component of the answer is the familiar section formula for that coordinate.