Geometric form of the scalar product?

\mathbf a\cdot\mathbf b=|\mathbf a||\mathbf b|\cos\theta

Formula for the angle between two vectors?

\cos\theta=\frac{\mathbf a\cdot\mathbf b}{|\mathbf a||\mathbf b|}

How do you find the angle between two lines?

Use their direction vectors: \cos\theta=\frac{|\mathbf d1\cdot\mathbf d2|}{|\mathbf d1||\mathbf d2|}.

Why use the modulus when finding the angle between lines?

The direction vectors may be opposite, giving an obtuse angle; the angle between lines is the acute one.

How do you find an unknown k in a perpendicular pair?

Set the scalar product to 0 and solve.

How do you show a triangle has a right angle at P?

Show \overrightarrow{PQ}\cdot\overrightarrow{PR}=0.

Scalar product and perpendicular vectorsAQA A-Level Further Maths: Flashcards

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Question

Scalar product of $\mathbf a=(a_1,a_2,a_3)$ and $\mathbf b=(b_1,b_2,b_3)$?

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Scalar product of $\mathbf a=(a_1,a_2,a_3)$ and $\mathbf b=(b_1,b_2,b_3)$ ?: $a_1b_1+a_2b_2+a_3b_3$
Geometric form of the scalar product?: $\mathbf a\cdot\mathbf b=|\mathbf a||\mathbf b|\cos\theta$
Formula for the angle between two vectors?: $\cos\theta=\frac{\mathbf a\cdot\mathbf b}{|\mathbf a||\mathbf b|}$
Is the scalar product a vector or a number?: A number (a scalar).
Condition for two non-zero vectors to be perpendicular?: $\mathbf a\cdot\mathbf b=0$
What is $\mathbf a\cdot\mathbf a$ ?: $|\mathbf a|^2$
What does $\mathbf a\cdot\mathbf b<0$ tell you about the angle?: It is obtuse.
$(2,-1,3)\cdot(4,5,1)$ ?: $8-5+3=6$
How do you find the angle between two lines?: Use their direction vectors: $\cos\theta=\frac{|\mathbf d_1\cdot\mathbf d_2|}{|\mathbf d_1||\mathbf d_2|}$ .
Why use the modulus when finding the angle between lines?: The direction vectors may be opposite, giving an obtuse angle; the angle between lines is the acute one.
How do you find an unknown $k$ in a perpendicular pair?: Set the scalar product to $0$ and solve.
How do you show a triangle has a right angle at $P$ ?: Show $\overrightarrow{PQ}\cdot\overrightarrow{PR}=0$ .

Exam questions on Scalar product and perpendicular vectors

The vectors $\mathbf a=\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ and $\mathbf b=\begin{pmatrix} 4 \\ 5 \\ 1 \end{pmatrix}$ are given.
Find the angle between $\mathbf a$ and $\mathbf b$ , giving your answer to the nearest $0.1^\circ$ .2 marks
Triangle $PQR$ has vertices $P(1,2,3)$ , $Q(4,0,5)$ and $R(3,4,2)$ .
Hence find the exact area of triangle $PQR$ .2 marks
The lines $l_1$ and $l_2$ have equations $\mathbf r=\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}+\lambda\begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}$ and $\mathbf r=\begin{pmatrix} 3 \\ -1 \\ 0 \end{pmatrix}+\mu\begin{pmatrix} 1 \\ 4 \\ 8 \end{pmatrix}$ .
Find the acute angle between $l_1$ and $l_2$ , to the nearest $0.1^\circ$ .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).