Unit vector in the direction of \mathbf{a}?

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

What is \overrightarrow{AB} in terms of position vectors?

\overrightarrow{AB}=\mathbf{b}-\mathbf{a}

Distance between (x1,y1,z1) and (x2,y2,z2)?

\sqrt{(x1-x2)^2+(y1-y2)^2+(z1-z2)^2}

When are two vectors parallel?

When one is a scalar multiple of the other, \mathbf{b}=\lambda\mathbf{a}.

What does a negative scalar multiple do?

Reverses the direction of the vector.

\overrightarrow{OA}+\overrightarrow{AB}=\overrightarrow{OB}

In parallelogram OABC, what is \overrightarrow{OB}?

\overrightarrow{OA}+\overrightarrow{OC}

If B is the midpoint of AC, find \mathbf{c}.

\mathbf{c}=2\mathbf{b}-\mathbf{a}

How do you prove a quadrilateral is a parallelogram with vectors?

Show one pair of opposite sides are equal vectors (equal in length and parallel).

Vectors in two and three dimensionsEdexcel International A Level Maths: Flashcards

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Magnitude of $\mathbf{a}=a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}$?

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Magnitude of $\mathbf{a}=a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}$ ?: $|\mathbf{a}|=\sqrt{a_1^2+a_2^2+a_3^2}$
Unit vector in the direction of $\mathbf{a}$ ?: $\hat{\mathbf{a}}=\frac{\mathbf{a}}{|\mathbf{a}|}$
What is $\overrightarrow{AB}$ in terms of position vectors?: $\overrightarrow{AB}=\mathbf{b}-\mathbf{a}$
Position vector of the midpoint of $AB$ ?: $\frac12(\mathbf{a}+\mathbf{b})$
Distance between $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ ?: $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}$
When are two vectors parallel?: When one is a scalar multiple of the other, $\mathbf{b}=\lambda\mathbf{a}$ .
What does a negative scalar multiple do?: Reverses the direction of the vector.
Triangle law?: $\overrightarrow{OA}+\overrightarrow{AB}=\overrightarrow{OB}$
In parallelogram $OABC$ , what is $\overrightarrow{OB}$ ?: $\overrightarrow{OA}+\overrightarrow{OC}$
If $B$ is the midpoint of $AC$ , find $\mathbf{c}$ .: $\mathbf{c}=2\mathbf{b}-\mathbf{a}$
$|2\mathbf{i}+3\mathbf{j}-6\mathbf{k}|$ ?: $\sqrt{4+9+36}=7$
How do you prove a quadrilateral is a parallelogram with vectors?: Show one pair of opposite sides are equal vectors (equal in length and parallel).

Exam questions on Vectors in two and three dimensions

The points $A$ and $B$ have position vectors $\mathbf{a}=2\mathbf{i}-\mathbf{j}+3\mathbf{k}$ and $\mathbf{b}=5\mathbf{i}+3\mathbf{j}-\mathbf{k}$ relative to the origin $O$ .
Find a unit vector in the direction of $\overrightarrow{AB}$ .2 marks
The points $A$ and $B$ have coordinates $(-1,2,3)$ and $(3,-2,5)$ relative to the origin $O$ , and $M$ is the midpoint of $AB$ .
The point $C$ is such that $B$ is the midpoint of $AC$ . Find the position vector of $C$ .2 marks
Triangle $ABC$ has vertices $A(1,2,-1)$ , $B(4,-2,3)$ and $C(5,5,3)$ , where the coordinates are in metres. A calculator may be used.
Show that triangle $ABC$ is isosceles.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).