How do you find the unit vector \hat{\mathbf{a}}?

Divide \mathbf{a} by its magnitude: \hat{\mathbf{a}}=\frac{\mathbf{a}}{|\mathbf{a}|}.

A vector has magnitude r and direction \theta. Write it in component form.

r\cos\theta\,\mathbf{i}+r\sin\theta\,\mathbf{j}.

How is the direction of x\mathbf{i}+y\mathbf{j} found?

\tan\theta=\frac{y}{x}, adjusting for the quadrant, measured anticlockwise from the positive x-axis.

State the triangle law.

\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}: place vectors head to tail.

State the parallelogram law.

The sum of two vectors drawn from the same point is the diagonal of the parallelogram they form.

How are two vectors shown to be parallel?

One is a scalar multiple of the other: \mathbf{b}=\lambda\mathbf{a}.

What does a negative scalar do to a vector?

It reverses the direction and scales the magnitude by |\lambda|.

How are \mathbf{a}-\mathbf{b} and \overrightarrow{BA} related to addition?

\mathbf{a}-\mathbf{b}=\mathbf{a}+(-\mathbf{b}) and \overrightarrow{BA}=-\overrightarrow{AB}.

If x1\mathbf{i}+y1\mathbf{j}=x2\mathbf{i}+y2\mathbf{j}, what follows?

x1=x2 and y1=y2: equate components.

What is the difference between a vector and a scalar?

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

Vectors in two dimensionsEdexcel A-Level Maths: Flashcards

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What is the magnitude of $x\mathbf{i}+y\mathbf{j}$?

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What is the magnitude of $x\mathbf{i}+y\mathbf{j}$ ?: $\sqrt{x^2+y^2}$ .
How do you find the unit vector $\hat{\mathbf{a}}$ ?: Divide $\mathbf{a}$ by its magnitude: $\hat{\mathbf{a}}=\frac{\mathbf{a}}{|\mathbf{a}|}$ .
A vector has magnitude $r$ and direction $\theta$ . Write it in component form.: $r\cos\theta\,\mathbf{i}+r\sin\theta\,\mathbf{j}$ .
How is the direction of $x\mathbf{i}+y\mathbf{j}$ found?: $\tan\theta=\frac{y}{x}$ , adjusting for the quadrant, measured anticlockwise from the positive $x$ -axis.
State the triangle law.: $\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}$ : place vectors head to tail.
State the parallelogram law.: The sum of two vectors drawn from the same point is the diagonal of the parallelogram they form.
How are two vectors shown to be parallel?: One is a scalar multiple of the other: $\mathbf{b}=\lambda\mathbf{a}$ .
What does a negative scalar do to a vector?: It reverses the direction and scales the magnitude by $|\lambda|$ .
How are $\mathbf{a}-\mathbf{b}$ and $\overrightarrow{BA}$ related to addition?: $\mathbf{a}-\mathbf{b}=\mathbf{a}+(-\mathbf{b})$ and $\overrightarrow{BA}=-\overrightarrow{AB}$ .
What is the magnitude of a unit vector?: $1$ .
If $x_1\mathbf{i}+y_1\mathbf{j}=x_2\mathbf{i}+y_2\mathbf{j}$ , what follows?: $x_1=x_2$ and $y_1=y_2$ : equate components.
What is the difference between a vector and a scalar?: A vector has magnitude and direction; a scalar has magnitude only.

Exam questions on Vectors in two dimensions

The vectors $\mathbf{a}$ and $\mathbf{b}$ are given by $\mathbf{a}=5\mathbf{i}-12\mathbf{j}$ and $\mathbf{b}=-\mathbf{i}+3\mathbf{j}$ .
Find the exact value of $|\mathbf{a}+2\mathbf{b}|$ .2 marks
The vector $\mathbf{p}$ is given by $\mathbf{p}=-4\mathbf{i}+4\sqrt{3}\,\mathbf{j}$ .
Find the vector of magnitude $24$ in the same direction as $\mathbf{p}$ , giving your answer in terms of $\mathbf{i}$ and $\mathbf{j}$ .2 marks
The vectors $\mathbf{a}$ and $\mathbf{b}$ are given by $\mathbf{a}=2\mathbf{i}+3\mathbf{j}$ and $\mathbf{b}=(k+1)\mathbf{i}+(k-2)\mathbf{j}$ , where $k$ is a constant.
Given that $\mathbf{b}$ is parallel to $\mathbf{a}$ , find the value of $k$ .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).