Cross Product Formula

The cross product or vector product is a binary operation on two vectors in three-dimensional space (R3) and is denoted by the symbol x. Two linearly independent vectors a and b, the cross product, a x b, is a vector that is perpendicular to both a and b and therefore normal to the plane containing them. 

Cross Product FormulaCross Product is given by,

\[\LARGE A\times B=\begin{vmatrix} i & j & k\\ a_{1} & a_{2}& a_{3} \\ b_{1} & b_{2}& b_{3} \end{vmatrix}\]

Where,
a1, a2, a3 are the components of the vector $\overrightarrow{a}$  and b1, b2 and b3 are the components of $\overrightarrow{b}$ .

Cross Product Formula is given by,

\[\LARGE a\times b=\left | a \right |\left | b \right |\sin \theta\]

Cross product formula is used to determine the cross product or angle between any two vectors based on the given problem.

Solved Examples

Question 1:Calculate the cross products of vectors a = <3, 4, 7> and b = <4, 9, 2>.

Solution:

The given vectors are, a = (3, 4, 7) and b = (4, 9, 2)
The cross product is given by
a $\times$ b =$\begin{vmatrix} i & j & k\\ a_{1} & a_{2}& a_{3} \\ b_{1} & b_{2}& b_{3} \end{vmatrix}$

a $\times$ b = $\begin{vmatrix} i & j & k \\3 & 4 & 7 \\4 & 9 & 2 \end{vmatrix}$

a $\times$ b = $i(4\times 2-9\times 7)-j(3 \times 2 – 4\times 7)+k(3\times 9-4\times 9)$

a $\times$ b = $i(8-63)-j(6-28)+k(27-36)$

a $\times$ b = $-55i+22j-9k$


Practise This Question

In California, citrus orchards were badly damaged by scale insects. To control these scale insects, ladybird beetles were released into these orchards. This type of pest control is known as