How to Find Cross Product of Vectors - Tutorial

How to Find Cross Product of Vectors - Tutorial, Formula, Example

Definition:

Cross product, is a vector product since it yields another vector rather than a scalar. As with the dot product, the cross product of two vectors contains valuable information about the two vectors themselves.

Formula :
The cross product of two vectors A= [ a1, a2, a3 ] and B= [ b1, b2, b3 ] is given by →     → a  x  b  = [ a2 b3 - a3 b2 , a2 b3 - a3 b1 , a1 b3 - a2 b1 ]

We may now write the formula for the cross product as,

An equivalent definition of the cross product is,

Example 1:

Find the cross product of a = 3i - 2j - 2k and b = -1i + 5k.

Given,

a = 3i - 2j - 2k b = -1i + 5k

Solution:

Example 2 :

Calculate the area of parallelogram spanned by the vectors a = 3i-3j+1k and b = 4i+9j+2k.

Given,

a = 3i-3j+1k b = 4i+9j+2k

Solution:

| a x b | = i( (-3).(2) - (1).(9)) - j((3).(2)-(1).(4)) + k((3).(9) + (3).(4)) | a x b | = -15 i - 2 j + 39 k The area is | a x b | Using the above expression for the cross product, we find that the area is, Area = √ (-15)2 + (-2)2 + 39 2 Area = 5 √ 70

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