Introducing vectors

Vectors as directed line segments, magnitude, equal, opposite, parallel and zero vectors, adding vectors head to tail and by the parallelogram rule, subtraction, and scalar multiples.

Vectors and their size

A vector has a magnitude (size) and a direction. A quantity with only a size, like mass or temperature, is a scalar. The directed line segment from $A$ to $B$ represents the vector $\overrightarrow{AB}$, and its length $|\overrightarrow{AB}|$ is the magnitude.

Equal vectors	Same magnitude and same direction, wherever they are drawn.
Opposite vector $-\underset{\sim}{a}$	Same magnitude, opposite direction. $\overrightarrow{BA} = -\overrightarrow{AB}$.
Zero vector $\underset{\sim}{0}$	Magnitude $0$, with no direction. $\overrightarrow{AA} = \underset{\sim}{0}$.
Parallel vectors	$\underset{\sim}{a} = \lambda\underset{\sim}{b}$ for some non-zero scalar $\lambda$.

Adding and subtracting

Subtraction is adding the opposite: $\underset{\sim}{a} - \underset{\sim}{b} = \underset{\sim}{a} + (-\underset{\sim}{b})$. Addition is commutative and associative, and $\underset{\sim}{a} + (-\underset{\sim}{a}) = \underset{\sim}{0}$.

Example 1●

A walker goes $3$ km east then $4$ km north. Find the resultant displacement.

Magnitude: $\sqrt{3^2 + 4^2} = 5$ km.

Direction: if $\theta$ is the angle east of north, $\tan\theta = \tfrac34$, so $\theta \approx 37^\circ$. The resultant is $5$ km on a bearing of about $037^\circ$T.

Example 2●●

$OABC$ is a parallelogram with $\overrightarrow{OA} = \underset{\sim}{a}$ and $\overrightarrow{OC} = \underset{\sim}{c}$. Express the diagonals $\overrightarrow{OB}$ and $\overrightarrow{AC}$ in terms of $\underset{\sim}{a}$ and $\underset{\sim}{c}$.

$\overrightarrow{OB} = \overrightarrow{OA} + \overrightarrow{AB} = \underset{\sim}{a} + \underset{\sim}{c}$, since $\overrightarrow{AB} = \overrightarrow{OC}$.

$\overrightarrow{AC} = \overrightarrow{AO} + \overrightarrow{OC} = -\underset{\sim}{a} + \underset{\sim}{c} = \underset{\sim}{c} - \underset{\sim}{a}$.

Scalar multiples

For a scalar $\lambda$, the vector $\lambda\underset{\sim}{a}$ has magnitude $|\lambda|\,|\underset{\sim}{a}|$. It points the same way as $\underset{\sim}{a}$ if $\lambda \gt 0$ and the opposite way if $\lambda \lt 0$. Scalar multiples obey the usual laws, such as $\lambda(\underset{\sim}{a} + \underset{\sim}{b}) = \lambda\underset{\sim}{a} + \lambda\underset{\sim}{b}$.

Example 3●●

$M$ is the midpoint of $AB$. Show that $\overrightarrow{OM} = \tfrac12(\underset{\sim}{a} + \underset{\sim}{b})$, where $\underset{\sim}{a} = \overrightarrow{OA}$ and $\underset{\sim}{b} = \overrightarrow{OB}$.

$\overrightarrow{OM} = \underset{\sim}{a} + \tfrac12\overrightarrow{AB} = \underset{\sim}{a} + \tfrac12(\underset{\sim}{b} - \underset{\sim}{a}) = \tfrac12(\underset{\sim}{a} + \underset{\sim}{b})$.

Flashcards

Vector or scalar?

A vector has magnitude and direction. A scalar has magnitude only.

$overrightarrow{AB} + overrightarrow{BC}$

$\overrightarrow{AC}$

$overrightarrow{BA}$

$-\overrightarrow{AB}$

Magnitude of $lambdaunderset{sim}{a}$

$|\lambda|\,|\underset{\sim}{a}|$

Practice

A walker goes $3$ km east then $4$ km north. Find the distance from the start, in km.
A boat sails $5$ km south then $12$ km west. Find the magnitude of its displacement, in km.
$|\underset{\sim}{a}| = 4$. Find $|-3\underset{\sim}{a}|$.
$\overrightarrow{PQ} + \overrightarrow{QR} + \overrightarrow{RS} = $
In parallelogram $OABC$ with $\overrightarrow{OA} = \underset{\sim}{a}$ and $\overrightarrow{OC} = \underset{\sim}{c}$, $\overrightarrow{CA} = $
$\overrightarrow{AB} + \overrightarrow{BA} = $
Which is a scalar?
$\underset{\sim}{u} = -2\underset{\sim}{v}$. Then $\underset{\sim}{u}$ and $\underset{\sim}{v}$ are:
$M$ is the midpoint of $AB$. Then $\overrightarrow{OM} = $