Introduction to Vectors

Starting point: what you already know

• Pythagoras and SOH CAH TOA.
• Coordinates: the point $(3,4)$ is 3 right and 4 up from the origin.
• True bearings and the difference between distance (how far) and displacement (how far and which way), from science.
• Gradient: lines with the same gradient are parallel.

Scalars and vectors

A scalar has size only: mass 5 kg, time 10 s, temperature $20^{\circ }\text{C}$.

A vector has size and direction: a displacement of 5 km north, a velocity of 30 m/s east, a force of 10 N downward.

We draw a vector as an arrow. Its length shows the size, and the arrowhead shows the direction.

Writing vectors

• As a directed line segment from $A$ to $B$: $\overrightarrow{AB}$.
• As a bold letter: $\mathbf{a}$. By hand, write $\underset{\sim }{a}$ with a squiggle under it.
• In component form: $\mathbf{a}=\left(\begin{array}{c}3\\ 4\end{array}\right)$ or $\mathbf{a}=3\mathbf{i}+4\mathbf{j}$.

Here $\mathbf{i}=\left(\begin{array}{c}1\\ 0\end{array}\right)$ is one unit right and $\mathbf{j}=\left(\begin{array}{c}0\\ 1\end{array}\right)$ is one unit up.

For points $A(x_1,y_1)$ and $B(x_2,y_2)$: $$\overrightarrow{AB}=\left(\begin{array}{c}{x}_2-x_1\\ y_2-y_1\end{array}\right)$$ "Where you finish minus where you start."

Magnitude and direction

The magnitude (length) of $\mathbf{a}=x\mathbf{i}+y\mathbf{j}$ is

$$|\mathbf{a}|=\sqrt{x^2+y^2}$$

and its direction angle $\theta$ from the positive x-axis satisfies $\tan \theta =\frac{y}{x}$. Check the quadrant from a sketch.

Adding and subtracting

Add vectors head to tail: start the second where the first ends. In components, just add.

$$\left(\begin{array}{c}{a}_1\\ a_2\end{array}\right)+\left(\begin{array}{c}{b}_1\\ b_2\end{array}\right)=\left(\begin{array}{c}{a}_1+b_1\\ a_2+b_2\end{array}\right)$$

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Subtract: $\mathbf{a}-\mathbf{b}=\mathbf{a}+(-\mathbf{b})$, where $-\mathbf{b}$ has the same length as $\mathbf{b}$ but points the opposite way.

Scalar multiples and unit vectors

$k\mathbf{a}$ stretches $\mathbf{a}$ by a factor $k$. If $k<0$ it also flips the direction. Two vectors are parallel when one is a scalar multiple of the other.

A unit vector has length 1. The unit vector in the direction of $\mathbf{a}$ is

$$\hat{\mathbf{a}}=\frac{\mathbf{a}}{|\mathbf{a}|}$$

Quick summary

• A vector has magnitude and direction. Write it as $\overrightarrow{AB}$, $\mathbf{a}$, a column, or $x\mathbf{i}+y\mathbf{j}$.
• $\overrightarrow{AB}$ = end point minus start point.
• $|x\mathbf{i}+y\mathbf{j}|=\sqrt{x^2+y^2}$.
• Add and subtract component by component. $k\mathbf{a}$ is parallel to $\mathbf{a}$.
• Unit vector: $\hat{\mathbf{a}}=\frac{\mathbf{a}}{|\mathbf{a}|}$.

Quiz