How do you find \overrightarrow{AB} from \mathbf{a} and \mathbf{b}?

\overrightarrow{AB}=\mathbf{b}-\mathbf{a} (end minus start).

What is the position vector of the midpoint of AB?

\frac12(\mathbf{a}+\mathbf{b}).

How are \overrightarrow{AB} and \overrightarrow{BA} related?

\overrightarrow{BA}=-\overrightarrow{AB}.

How do you show P, Q, R are collinear?

Show \overrightarrow{PQ} is a scalar multiple of \overrightarrow{QR}; they are parallel and share the point Q.

How do you show a triangle is isosceles?

Find the three side lengths and show two are equal.

How do you show a triangle is right-angled?

Show AB^2+AC^2=BC^2 for the squared side lengths (converse of Pythagoras).

In parallelogram ABDC, what is \overrightarrow{BD}?

\overrightarrow{BD}=\overrightarrow{AC}, so \mathbf{d}=\mathbf{b}+\mathbf{c}-\mathbf{a}.

What do the diagonals of a parallelogram do?

They bisect each other: they share the same midpoint.

Why work with squared lengths when comparing sides?

It avoids surds and the comparison is exactly the same.

Are the coordinates of A(3,-2) linked to \mathbf{a}?

Yes: \mathbf{a}=3\mathbf{i}-2\mathbf{j}, the coordinates are its components.

Position vectors and distanceEdexcel A-Level Maths: Flashcards

Card 1 of 120 of 12 known

Question

What is a position vector?

Tap or press Space to reveal

Tap card or press Space to flip

See all 12 cards

What is a position vector?: The vector $\overrightarrow{OA}$ from the origin $O$ to the point $A$ , written $\mathbf{a}$ .
How do you find $\overrightarrow{AB}$ from $\mathbf{a}$ and $\mathbf{b}$ ?: $\overrightarrow{AB}=\mathbf{b}-\mathbf{a}$ (end minus start).
What is the position vector of the midpoint of $AB$ ?: $\frac12(\mathbf{a}+\mathbf{b})$ .
State the distance formula for $(x_1,y_1)$ and $(x_2,y_2)$ .: $d^2=(x_1-x_2)^2+(y_1-y_2)^2$ .
How are $\overrightarrow{AB}$ and $\overrightarrow{BA}$ related?: $\overrightarrow{BA}=-\overrightarrow{AB}$ .
How do you show $P$ , $Q$ , $R$ are collinear?: Show $\overrightarrow{PQ}$ is a scalar multiple of $\overrightarrow{QR}$ ; they are parallel and share the point $Q$ .
How do you show a triangle is isosceles?: Find the three side lengths and show two are equal.
How do you show a triangle is right-angled?: Show $AB^2+AC^2=BC^2$ for the squared side lengths (converse of Pythagoras).
In parallelogram $ABDC$ , what is $\overrightarrow{BD}$ ?: $\overrightarrow{BD}=\overrightarrow{AC}$ , so $\mathbf{d}=\mathbf{b}+\mathbf{c}-\mathbf{a}$ .
What do the diagonals of a parallelogram do?: They bisect each other: they share the same midpoint.
Why work with squared lengths when comparing sides?: It avoids surds and the comparison is exactly the same.
Are the coordinates of $A(3,-2)$ linked to $\mathbf{a}$ ?: Yes: $\mathbf{a}=3\mathbf{i}-2\mathbf{j}$ , the coordinates are its components.

Exam questions on Position vectors and distance

Relative to an origin $O$ , the points $A$ and $B$ have position vectors $\mathbf{a}=2\mathbf{i}+7\mathbf{j}$ and $\mathbf{b}=8\mathbf{i}-\mathbf{j}$ .
The point $M$ is the midpoint of $AB$ . Find the position vector of $M$ .2 marks
Relative to an origin $O$ , the points $P$ , $Q$ and $R$ have position vectors $\mathbf{p}=\mathbf{i}+2\mathbf{j}$ , $\mathbf{q}=5\mathbf{i}+5\mathbf{j}$ and $\mathbf{r}=13\mathbf{i}+11\mathbf{j}$ .
Find the exact distance $PR$ .2 marks
Relative to an origin $O$ , the points $A$ , $B$ and $C$ are the vertices of a triangle, with position vectors $\mathbf{a}=\mathbf{i}+2\mathbf{j}$ , $\mathbf{b}=7\mathbf{i}+4\mathbf{j}$ and $\mathbf{c}=3\mathbf{i}+8\mathbf{j}$ .
Show that triangle $ABC$ is isosceles.3 marks

See the full worksheet

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).