Direction of \mathbf{a}\times\mathbf{b}?

Perpendicular to both \mathbf{a} and \mathbf{b}, by the right-hand rule.

When is \mathbf{a}\times\mathbf{b}=\mathbf{0}?

When \mathbf{a} and \mathbf{b} are parallel (or one is zero).

How do you calculate \mathbf{a}\times\mathbf{b} in components?

Expand \begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\ a1&a2&a3\\ b1&b2&b3\end{vmatrix} along the top row.

What does |\mathbf{a}\times\mathbf{b}| represent?

The area of the parallelogram with sides \mathbf{a} and \mathbf{b}.

Area of a triangle with sides \mathbf{a} and \mathbf{b}?

\frac12|\mathbf{a}\times\mathbf{b}|

Distance from C to line AB?

\frac{\left|\overrightarrow{AB}\times\overrightarrow{AC}\right|}{|\overrightarrow{AB}|}

Scalar triple product as a determinant?

The determinant with rows \mathbf{a}, \mathbf{b}, \mathbf{c}.

What does \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})=0 tell you?

The three vectors are coplanar.

Volume of a parallelepiped and of a tetrahedron?

|\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})| and \frac16|\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})|

The vector productEdexcel International A Level Further Maths: Flashcards

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Magnitude of $\mathbf{a}\times\mathbf{b}$?

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Magnitude of $\mathbf{a}\times\mathbf{b}$ ?: $|\mathbf{a}||\mathbf{b}|\sin\theta$
Direction of $\mathbf{a}\times\mathbf{b}$ ?: Perpendicular to both $\mathbf{a}$ and $\mathbf{b}$ , by the right-hand rule.
What is $\mathbf{b}\times\mathbf{a}$ in terms of $\mathbf{a}\times\mathbf{b}$ ?: $-\mathbf{a}\times\mathbf{b}$
When is $\mathbf{a}\times\mathbf{b}=\mathbf{0}$ ?: When $\mathbf{a}$ and $\mathbf{b}$ are parallel (or one is zero).
What is $\mathbf{a}\times\mathbf{a}$ ?: $\mathbf{0}$
How do you calculate $\mathbf{a}\times\mathbf{b}$ in components?: Expand $\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\ a_1&a_2&a_3\\ b_1&b_2&b_3\end{vmatrix}$ along the top row.
What does $|\mathbf{a}\times\mathbf{b}|$ represent?: The area of the parallelogram with sides $\mathbf{a}$ and $\mathbf{b}$ .
Area of a triangle with sides $\mathbf{a}$ and $\mathbf{b}$ ?: $\frac12|\mathbf{a}\times\mathbf{b}|$
Distance from $C$ to line $AB$ ?: $\frac{\left|\overrightarrow{AB}\times\overrightarrow{AC}\right|}{|\overrightarrow{AB}|}$
Scalar triple product as a determinant?: The determinant with rows $\mathbf{a}$ , $\mathbf{b}$ , $\mathbf{c}$ .
What does $\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})=0$ tell you?: The three vectors are coplanar.
Volume of a parallelepiped and of a tetrahedron?: $|\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})|$ and $\frac16|\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})|$

Exam questions on The vector product

The vectors $\mathbf{a}=2\mathbf{i}+\mathbf{j}-\mathbf{k}$ and $\mathbf{b}=\mathbf{i}+3\mathbf{j}+2\mathbf{k}$ are given.
Find the area of the triangle with sides $\mathbf{a}$ and $\mathbf{b}$ , and write down a vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$ with magnitude $1$ .2 marks
A parallelepiped has edges represented by the vectors $\mathbf{a}=\mathbf{i}+\mathbf{j}+2\mathbf{k}$ , $\mathbf{b}=3\mathbf{i}+\mathbf{j}$ and $\mathbf{c}=2\mathbf{j}+\mathbf{k}$ .
The vector $\mathbf{d}=2\mathbf{j}+\lambda\mathbf{k}$ is such that $\mathbf{a}$ , $\mathbf{b}$ and $\mathbf{d}$ lie in the same plane. Find the value of $\lambda$ .2 marks
The points $P$ , $Q$ , $R$ and $S$ have coordinates $(1,2,0)$ , $(3,3,2)$ , $(2,5,1)$ and $(0,1,4)$ respectively.
Find the exact area of triangle $PQR$ .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).