What is the magnitude of \mathbf{a}\times\mathbf{b}?

|\mathbf{a}||\mathbf{b}|\sin\theta.

What is the direction of \mathbf{a}\times\mathbf{b}?

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

How are \mathbf{a}\times\mathbf{b} and \mathbf{b}\times\mathbf{a} related?

\mathbf{b}\times\mathbf{a}=-\mathbf{a}\times\mathbf{b}.

How do you find a vector perpendicular to two given vectors?

Take their vector product; any non-zero multiple also works.

What area does |\mathbf{a}\times\mathbf{b}| give?

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

State the area of a triangle with sides \mathbf{a} and \mathbf{b} from a common vertex.

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

Define the scalar triple product.

\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}): a number, found by taking the cross product first.

State the cyclic property of the scalar triple product.

\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})=\mathbf{b}\cdot(\mathbf{c}\times\mathbf{a})=\mathbf{c}\cdot(\mathbf{a}\times\mathbf{b}).

State the volume of a parallelepiped with edges \mathbf{a}, \mathbf{b}, \mathbf{c}.

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

State the volume of a tetrahedron with edges \mathbf{a}, \mathbf{b}, \mathbf{c}.

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

Vector product and scalar triple productEdexcel A-Level Further Maths: Flashcards

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State the component form of $\mathbf{a}\times\mathbf{b}$.

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State the component form of $\mathbf{a}\times\mathbf{b}$ .: $(a_2b_3-a_3b_2)\mathbf{i}-(a_1b_3-a_3b_1)\mathbf{j}+(a_1b_2-a_2b_1)\mathbf{k}$ .
What is the magnitude of $\mathbf{a}\times\mathbf{b}$ ?: $|\mathbf{a}||\mathbf{b}|\sin\theta$ .
What is the direction of $\mathbf{a}\times\mathbf{b}$ ?: Perpendicular to both $\mathbf{a}$ and $\mathbf{b}$ , given by the right-hand rule.
How are $\mathbf{a}\times\mathbf{b}$ and $\mathbf{b}\times\mathbf{a}$ related?: $\mathbf{b}\times\mathbf{a}=-\mathbf{a}\times\mathbf{b}$ .
What is $\mathbf{a}\times\mathbf{a}$ ?: The zero vector $\mathbf{0}$ .
What is $\mathbf{i}\times\mathbf{j}$ ?: $\mathbf{k}$ .
How do you find a vector perpendicular to two given vectors?: Take their vector product; any non-zero multiple also works.
What area does $|\mathbf{a}\times\mathbf{b}|$ give?: The area of the parallelogram with adjacent sides $\mathbf{a}$ and $\mathbf{b}$ .
State the area of a triangle with sides $\mathbf{a}$ and $\mathbf{b}$ from a common vertex.: $\frac12|\mathbf{a}\times\mathbf{b}|$ .
Define the scalar triple product.: $\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})$ : a number, found by taking the cross product first.
State the cyclic property of the scalar triple product.: $\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})=\mathbf{b}\cdot(\mathbf{c}\times\mathbf{a})=\mathbf{c}\cdot(\mathbf{a}\times\mathbf{b})$ .
What happens to the triple product when two vectors are swapped?: It changes sign.
State the volume of a parallelepiped with edges $\mathbf{a}$ , $\mathbf{b}$ , $\mathbf{c}$ .: $|\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})|$ .
State the volume of a tetrahedron with edges $\mathbf{a}$ , $\mathbf{b}$ , $\mathbf{c}$ .: $\frac16|\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})|$ .

Exam questions on Vector product and scalar triple product

The vectors $\mathbf{a}$ and $\mathbf{b}$ are given by $\mathbf{a}=\mathbf{i}+2\mathbf{j}+3\mathbf{k}$ and $\mathbf{b}=2\mathbf{i}-\mathbf{j}+\mathbf{k}$ .
The points $O$ , $A$ and $B$ have position vectors $\mathbf{0}$ , $\mathbf{a}$ and $\mathbf{b}$ . Find the exact area of triangle $OAB$ .2 marks
The tetrahedron $OABC$ has $O$ at the origin, and $\overrightarrow{OA}=\mathbf{i}+2\mathbf{j}+\mathbf{k}$ , $\overrightarrow{OB}=\mathbf{j}+3\mathbf{k}$ and $\overrightarrow{OC}=2\mathbf{i}+\mathbf{k}$ .
Find a vector that is perpendicular to both $\overrightarrow{OA}$ and $\overrightarrow{OB}$ .2 marks
The tetrahedron $ABCD$ has vertices $A(1,0,2)$ , $B(2,3,0)$ , $C(4,1,1)$ and $D(2,-1,4)$ .
Find the exact area of triangle $ABC$ .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).