What is a unit vector in the direction of a?

\frac{\mathbf a}{|\mathbf a|}, which has magnitude 1.

How is the direction of a vector usually measured?

Anticlockwise from the positive x-direction.

What does \tan^{-1}\frac yx give?

An angle in the first or fourth quadrant only, so adjust for the quadrant.

Direction of -5\mathbf i+12\mathbf j to 1 d.p.?

112.6^\circ (second quadrant: 180^\circ-67.4^\circ)

Components of magnitude r at angle \theta?

r\cos\theta\,\mathbf i+r\sin\theta\,\mathbf j

What is a position vector?

\overrightarrow{OA}, the vector from the origin to A; its components are the coordinates of A.

\overrightarrow{AB} in terms of position vectors?

\mathbf b-\mathbf a (end minus start)

How do you find the distance between two points from their position vectors?

Find \mathbf b-\mathbf a then its magnitude |\mathbf b-\mathbf a|.

What does AB^2+BC^2=AC^2 show?

A right angle at B (converse of Pythagoras).

Magnitude, direction and position vectorsAQA A-Level Maths: Flashcards

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Magnitude of $x\mathbf i+y\mathbf j$?

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Magnitude of $x\mathbf i+y\mathbf j$ ?: $\sqrt{x^2+y^2}$
What is a unit vector in the direction of a?: $\frac{\mathbf a}{|\mathbf a|}$ , which has magnitude $1$ .
Magnitude of $5\mathbf i-12\mathbf j$ ?: $13$
How is the direction of a vector usually measured?: Anticlockwise from the positive $x$ -direction.
What does $\tan^{-1}\frac yx$ give?: An angle in the first or fourth quadrant only, so adjust for the quadrant.
Direction of $-5\mathbf i+12\mathbf j$ to 1 d.p.?: $112.6^\circ$ (second quadrant: $180^\circ-67.4^\circ$ )
Components of magnitude $r$ at angle $\theta$ ?: $r\cos\theta\,\mathbf i+r\sin\theta\,\mathbf j$
What is a position vector?: $\overrightarrow{OA}$ , the vector from the origin to $A$ ; its components are the coordinates of $A$ .
$\overrightarrow{AB}$ in terms of position vectors?: $\mathbf b-\mathbf a$ (end minus start)
How do you find the distance between two points from their position vectors?: Find $\mathbf b-\mathbf a$ then its magnitude $|\mathbf b-\mathbf a|$ .
Convert magnitude $8$ , direction $150^\circ$ to components.: $-4\sqrt3\,\mathbf i+4\mathbf j$
What does $AB^2+BC^2=AC^2$ show?: A right angle at $B$ (converse of Pythagoras).

Exam questions on Magnitude, direction and position vectors

The vector $\mathbf p=5\mathbf i-12\mathbf j$ .
Find a unit vector in the direction of $\mathbf p$ .2 marks
The points $A$ and $B$ have position vectors $\mathbf a=2\mathbf i+3\mathbf j$ and $\mathbf b=8\mathbf i-5\mathbf j$ relative to the origin $O$ .
The point $C$ has position vector $3\mathbf a$ . Find the exact distance $AC$ .2 marks
The vector $\mathbf v$ has magnitude $8$ and direction $150^\circ$ , measured anticlockwise from the positive $x$ -direction. The vector $\mathbf w=4\sqrt3\,\mathbf i+3\mathbf j$ .
Express $\mathbf v$ in the form $x\mathbf i+y\mathbf j$ , giving exact values of $x$ and $y$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).