Cross Product/Vector Product of Vectors

Readers are already familiar with a three-dimensional right-handed rectangular coordinate system. In this system, a counterclockwise rotation of the x-axis into the positive y-axis indicates that a right-handed (standard) screw would advance in the direction of the positive z-axis as shown in the figure.

The vector product or cross product, of two vectors a and b with an angle α between them is mathematically calculated as

a × b = |a| |b| sin α

It is to be noted that the cross product is a vector with a specified direction. The resultant is always perpendicular to both a and b. 

Also, if given two vectors, [Tex]\mathbf{a} = (a_1, a_2, a_3)[/Tex] and [Tex]\mathbf{b} = (b_1, b_2, b_3)[/Tex], their cross product, denoted by a × b, is calculated as:

[Tex]\mathbf{a} \times \mathbf{b} = (a_2b_3 – a_3b_2, a_3b_1 – a_1b_3, a_1b_2 – a_2b_1)[/Tex]

In case a and b are parallel vectors, the resultant shall be zero as sin(0) = 0

Properties of Cross Product

• Cross Product generates a vector quantity. The resultant is always perpendicular to both a and b.
• Cross Product of parallel vectors/collinear vectors is zero as sin(0) = 0.

                 i × i = j × j = k × k = 0

• Cross product of two mutually perpendicular vectors with unit magnitude each is unity. (Since sin(0)=1)
• Cross product is not commutative.

                 a × b is not equal to b × a

• Cross product is distributive over addition

                 a × (b + c) = a × b+ a × c

• If k is a scalar then,

                 k(a × b) = k(a) × b = a × k(b)

• On moving in a clockwise direction and taking the cross product of any two pair of the unit vectors we get the third one and in an anticlockwise direction, we get the negative resultant.

The following results can be established:

Cross product in Determinant Form

If the vector a is represented as a = a1x + a2y + a3z and vector b is represented as b = b1x + b2y + b3z

Then the cross product a × b can be computed using determinant form

[Tex]\begin{array}{ccc} x & y & z \\ a 1 & a 2 & a 3 \\ b 1 & b 2 & b 3 \end{array} [/Tex]

Then, a × b = x(a₂b₃ – b₂a₃) + y(a₃b₁ – a₁b₃) + z(a₁b₂ – a₂b₁)

If a and b are the adjacent sides of the parallelogram OXYZ and α is the angle between the vectors a and b.

Then the area of the parallelogram is given by |a × b| = |a| |b|sin.α

Examples of Cross product of Vectors

Example 1. Find the cross product of two vectors a and b if their magnitudes are 5 and 10 respectively. Given that angle between then is 30°.

Solution:

a × b = a.b.sin (30) = (5) (10) (1/2) = 25 perpendicular to a and b

Example 2. Find the area of a parallelogram whose adjacent sides are  

a = 4i+2j -3k

b= 2 i +j-4k

Solution: 

The area is calculated by finding the cross product of adjacent sides

a × b = x(a₂b₃ – b₂a₃) + y(a₃b₁ – a₁b₃) + z(a₁b₂ – a₂b₁)

= i(-8+3) + j(-6+16) + k(4-4)

= -5i +10j

Therefore, the magnitude of area is [Tex]\sqrt{(5^2 +10^2)} [/Tex]

= [Tex]\sqrt{(25+100)} [/Tex]

= [Tex]\sqrt{(125)} =5\sqrt{5} [/Tex]

A quantity that is characterized not only by magnitude but also by its direction, is called a vector. Velocity, force, acceleration, momentum, etc. are vectors.  

Vectors can be multiplied in two ways:

• Scalar product or Dot product
• Vector Product or Cross product