Cross Product Formula

We can multiply the Vectors in two ways, one is the scalar product where the result is a scalar and the other is the cross or vector product where is the result is a vector. In this topic, we will look at the cross or the vector product of two vectors. The vector product or cross product is a binary type of operation between two vectors in a three-dimensional space. Thus the result is a vector perpendicular to the vectors that multiply, and therefore normal to the plane that contains them. The student will also learn the Cross product formula with examples. Let us learn it!

Cross Product Formula

What is the Concept of the Cross Product?

We have already studied the three-dimensional right-handed rectangular coordinate system. The cross product or vector product is a binary operation on two vectors in three-dimensional (3D) space. And it is represented by the symbol ×. Two linearly independent vectors a and b, the cross product, a × b, is a vector which is perpendicular to both vectors a and b. Also, it will be towards normal to the plane containing them.

Due to its ability to obtain a vector perpendicular to two other vectors, whose direction varies according to the angle formed between these two vectors, this operation is often applied to solve mathematical, physical or engineering problems.

Formula for Cross Product

Cross Product is: \(a\times b =\begin{vmatrix} i & j & k\\ a_{1} & a_{2}& a_{3} \\ b_{1} & b_{2}& b_{3} \end{vmatrix} \\\)

Where, \(a_{1}, a_{2}, a_{3}\) are the components of the vector \(\overrightarrow{a}  and b_{1} , b_{2} and b_{3}\) are the components of \(\overrightarrow{b}\)

Also,

\(\vec{a} \times \vec{b} = \vec{a}\vec{b}sin \theta \hat{n}\)

Where \(\theta\) is the angle between two given vectors \(\vec{a} and \vec{b}.\)

Also, \(\hat{n}\) is a unit vector.

In other words, the Cross Product Formula is as follows:

\(a \times b=\left | a \right |\left | b \right |\sin \theta\)

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

Some Important Points:

1. \(\vec {a} \times \vec {b}\) is a vector.
2. If either \(\vec {a} = 0 or \vec {b}\) = 0, then \(\theta\) is not defined and we may say, \(\vec {a} \times \vec {b} =  \vec {0}\)
3. A cross or vector product is not commutative in nature. This id due to the reason that, \(\vec {a} \times \vec {b} = −   \vec {b} \times \vec {a}\)

4. If \(\vec {a} and \vec {b}\) represent the two sides of a triangle, then its area will be \(\frac{1}{2} |\vec {a} \times \vec {b} |\)

Solved Examples for Cross Product Formula

Q.1: Calculate the cross products of vectors a = (3, 5, 7) and b = (5, 9, 2).

Solution: The given vectors are, a = (3, 5, 7) and b = (5, 9, 2)

The cross product is:

\(a \times b = \begin{vmatrix} i & j & k\\ a_{1} & a_{2}& a_{3} \\ b_{1} & b_{2}& b_{3} \end{vmatrix} \\\)

\(a \times b = \begin{vmatrix} i & j & k \\3 & 5 & 7 \\5 & 9 & 2 \end{vmatrix} \\\)

\(= i(5\times 2-9\times 7)-j(3 \times 2 – 5\times 7)+k(3\times 9-5\times 5) \\\)

=  i(10-63)-j(6-35)+k(27-25)

=  -53i + 29 j + 2k

Thus cross product will be (-53 , 29 , 2).