Why a Cricket Throw Needs Both a Speed and a Direction

Saying a fielder threw the ball at $25$ m/s is not enough to know where it lands — you also need its direction. Quantities with both size and direction are vectors, and they describe motion, force and position precisely.

This part covers magnitude and direction, types of vectors and position vectors, vector addition and scalar multiplication, and the section formula.

**A scalar has only magnitude, while a vector has magnitude and direction; for $\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$ the magnitude is $\sqrt{x^2+y^2+z^2}$, the direction cosines are $\frac{x}{r},\frac{y}{r},\frac{z}{r}$, and any numbers proportional to them are direction ratios.

Examples:

- Scalars** — mass $5$ kg, temperature $30^{\circ }$C, distance $2$ km
- Vectors — displacement $2$ km north, velocity $60$ km/h east, force $10$ N downward

Direction cosines. If $\vec{r}$ makes angles $\alpha$, $\beta$ and $\gamma$ with the $x$, $y$ and $z$ axes, then $l=\cos \alpha$, $m=\cos \beta$, $n=\cos \gamma$, and

$$l^2+m^2+n^2=1$$

Worked example. For $\vec{r}=2\hat{i}-\hat{j}+2\hat{k}$:

$$|\vec{r}|=\sqrt{4+1+4}=3l,m,n=\frac{2}{3},-\frac{1}{3},\frac{2}{3}$$

Check: $\frac{4}{9}+\frac{1}{9}+\frac{4}{9}=1$. One set of direction ratios is $2,-1,2$.

An everyday example. An auto driver told to go 3 km must ask which way — the distance is a scalar, while the trip with its direction is a vector.

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

**Equal vectors share magnitude and direction, a unit vector has magnitude $1$, the zero vector has magnitude $0$, collinear vectors are parallel to one line, and a negative vector has the same magnitude but opposite direction; the position vector of $P(x,y,z)$ is $x\hat{i}+y\hat{j}+z\hat{k}$.

Types:

- Zero vector** $\vec{0}$ — initial and terminal points coincide
- Unit vector — magnitude $1$; $\hat{a}=\frac{\vec{a}}{|\vec{a}|}$
- Equal vectors — same magnitude and direction, whatever their starting points
- Collinear vectors — parallel to the same line, in the same or opposite direction
- Negative vector — $-\vec{a}$ has the length of $\vec{a}$ but points the other way

Position vector and components. For $P(x,y,z)$, $\overrightarrow{OP}=x\hat{i}+y\hat{j}+z\hat{k}$; $x\hat{i}$, $y\hat{j}$, $z\hat{k}$ are its vector components and $x$, $y$, $z$ its scalar components. The vector from $A$ to $B$ is $\overrightarrow{AB}=\vec{b}-\vec{a}$.

Worked example. For $A(1,2,3)$ and $B(4,6,3)$:

$$\overrightarrow{AB}=3\hat{i}+4\hat{j},|\overrightarrow{AB}|=5,\text{unit vector}=\frac{3}{5}\hat{i}+\frac{4}{5}\hat{j}$$

An everyday example. Two cars moving at the same speed in the same direction on parallel lanes of a highway have equal velocity vectors, even though they are in different places.

The substance. Equal vectors need not start at the same point — only magnitude and direction must match.

**By the triangle law, placing the tail of $\vec{b}$ at the head of $\vec{a}$ makes $\vec{a}+\vec{b}$ the vector from the tail of $\vec{a}$ to the head of $\vec{b}$; the parallelogram law gives the same sum as a diagonal; and multiplying by a scalar $\lambda$ scales the length by $|\lambda |$, reversing the direction when $\lambda <0$.

Laws of addition:

- Triangle law** — $\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}$
- Parallelogram law — for co-initial sides $\vec{a}$ and $\vec{b}$, the diagonal from the common point is $\vec{a}+\vec{b}$

Properties:

- Commutative — $\vec{a}+\vec{b}=\vec{b}+\vec{a}$
- Associative — $(\vec{a}+\vec{b})+\vec{c}=\vec{a}+(\vec{b}+\vec{c})$
- Scalar multiplication — $|\lambda \vec{a}|=|\lambda ||\vec{a}|$ and $\lambda (\vec{a}+\vec{b})=\lambda \vec{a}+\lambda \vec{b}$

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

$$\vec{a}+\vec{b}=3\hat{i}-2\hat{j}+3\hat{k}2\vec{a}-\vec{b}=3\hat{i}+5\hat{j}-6\hat{k}$$

Collinearity. $\vec{a}$ and $\vec{b}$ are collinear exactly when $\vec{b}=\lambda \vec{a}$ for some scalar $\lambda$.

An everyday example. A ferry crossing a river is pushed across by its engine and carried downstream by the current; its actual path is the vector sum of the two velocities, the diagonal of a parallelogram.

The substance. Around a closed triangle, the vectors add to zero — $\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CA}=\vec{0}$.

**If $P$ divides the segment joining $A(\vec{a})$ and $B(\vec{b})$ internally in the ratio $m:n$, its position vector is $\frac{m\vec{b}+n\vec{a}}{m+n}$; externally it is $\frac{m\vec{b}-n\vec{a}}{m-n}$.

Special cases.** The midpoint is $\frac{\vec{a}+\vec{b}}{2}$, and the centroid of a triangle with vertices $\vec{a}$, $\vec{b}$, $\vec{c}$ is $\frac{\vec{a}+\vec{b}+\vec{c}}{3}$.

Worked example 1 — internal. $A$ and $B$ have position vectors $\hat{i}+2\hat{j}-\hat{k}$ and $4\hat{i}-\hat{j}+5\hat{k}$. The point dividing $AB$ internally in the ratio $2:1$ is

$$\frac{2(4\hat{i}-\hat{j}+5\hat{k})+(\hat{i}+2\hat{j}-\hat{k})}{3}=\frac{9\hat{i}+9\hat{k}}{3}=3\hat{i}+3\hat{k}$$

Worked example 2 — external. The same ratio $2:1$ externally:

$$\frac{2(4\hat{i}-\hat{j}+5\hat{k})-(\hat{i}+2\hat{j}-\hat{k})}{2-1}=7\hat{i}-4\hat{j}+11\hat{k}$$

An everyday example. Marking a point two-thirds of the way along a straight road between two villages is an internal division in the ratio $2:1$.

The substance. **In the formula, $m$ multiplies the far end $\vec{b}$** — swapping $m$ and $n$ measures the point from the other end.