Vector (Cross) Product & Area

MATVector (Cross) Product

The vector (cross) product $\vec{a}\times\vec{b}$ produces a vector perpendicular to both $\vec{a}$ and $\vec{b}$ , with magnitude $|\vec{a}|\,|\vec{b}|\sin\theta$ equal to the area of the parallelogram they span. It gives unit normals, areas of triangles and parallelograms, the sine of the angle between vectors, and torque (moment of a force).

ISC tests it via determinant computation, area formulas, and physics applications.

Cross product (determinant form)

\vec{a}\times\vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}

\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}

\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}

Magnitude of cross product

|\vec{a}\times\vec{b}| = |\vec{a}|\,|\vec{b}|\sin\theta

\theta

is the angle between

\vec{a}

and

\vec{b}

0 \le \theta \le \pi

, so

\sin\theta \ge 0

Unit vector perpendicular to both

\hat{n} = \pm\,\dfrac{\vec{a}\times\vec{b}}{|\vec{a}\times\vec{b}|}

\hat{n}

is perpendicular to the plane of

\vec{a}

and

\vec{b}

; both signs are valid.

Area of parallelogram (adjacent sides)

\text{Area} = |\vec{a}\times\vec{b}|

\vec{a},\vec{b}

are the adjacent sides of the parallelogram.

Area of parallelogram (diagonals)

\text{Area} = \dfrac{1}{2}\,|\vec{d_1}\times\vec{d_2}|

\vec{d_1},\vec{d_2}

are the diagonals of the parallelogram.

Area of triangle and torque

\text{Area}_{\triangle} = \dfrac{1}{2}\,|\vec{AB}\times\vec{AC}|, \qquad \vec{\tau} = \vec{r}\times\vec{F}

\vec{AB},\vec{AC}

are two sides from a common vertex; torque

\vec{\tau}

uses position vector

\vec{r}

of the point of application and force

\vec{F}

The cross product is anti-commutative: $\vec{a}\times\vec{b} = -(\vec{b}\times\vec{a})$ .
If $\vec{a}\times\vec{b}=\vec{0}$ (with both nonzero), the vectors are parallel; in particular $\vec{a}\times\vec{a}=\vec{0}$ .
Basis cross products cycle: $\hat{i}\times\hat{j}=\hat{k}$ , $\hat{j}\times\hat{k}=\hat{i}$ , $\hat{k}\times\hat{i}=\hat{j}$ ; reversing order flips the sign.
Expand the determinant along the top row, watching the sign pattern $+,-,+$ on $\hat{i},\hat{j},\hat{k}$ .
For the area of a triangle with vertices $A,B,C$ , form two side vectors from a common vertex, then take half the magnitude of their cross product.
The sine of the angle is $\sin\theta = \dfrac{|\vec{a}\times\vec{b}|}{|\vec{a}|\,|\vec{b}|}$ , which is always non-negative since $0\le\theta\le\pi$ .
When the sides of a parallelogram are given, use $|\vec{a}\times\vec{b}|$ ; when the diagonals are given, use $\dfrac{1}{2}|\vec{d_1}\times\vec{d_2}|$ .

Where the marks go

Omitting the $\dfrac{1}{2}$ for triangle area, or omitting it for the diagonals form of parallelogram area.
Mishandling the middle $\hat{j}$ term of the determinant, whose cofactor carries a minus sign.
Giving the cross product as a scalar; it is a vector, while the dot product is the scalar one.
Forgetting to normalise (divide by the magnitude) when a unit perpendicular vector is required.

How the board asks it

Numericalthe cross product (determinant form) and unit normal
If $\vec{a}=2\hat{i}+\hat{j}+3\hat{k}$ and $\vec{b}=3\hat{i}+5\hat{j}-2\hat{k}$ , find $\vec{a}\times\vec{b}$ and hence a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$ .
Numericalarea of a triangle as half the magnitude of a cross product
Find the area of the triangle whose vertices are $A(1,1,2)$ , $B(2,3,5)$ and $C(1,5,5)$ .
Numericalsine of the angle from the magnitude of the cross product
If $\vec{a}=\hat{i}-7\hat{j}+7\hat{k}$ and $\vec{b}=3\hat{i}-2\hat{j}+2\hat{k}$ , find $|\vec{a}\times\vec{b}|$ and hence the sine of the angle between $\vec{a}$ and $\vec{b}$ .
Numericalarea of a parallelogram from its diagonals
Find the area of a parallelogram whose diagonals are $\vec{d_1}=3\hat{i}+\hat{j}-2\hat{k}$ and $\vec{d_2}=\hat{i}-3\hat{j}+4\hat{k}$ .
Applicationtorque as the moment of a force, $\vec{r}\times\vec{F}$
A force $\vec{F}=2\hat{i}+\hat{j}-\hat{k}$ acts at the point $A(1,2,1)$ . Find the moment of $\vec{F}$ about the point $B(2,0,3)$ .

Practise this topic

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Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.