Cross Product of Vectors

Vector product is otherwise known as cross product.  Cross product of two vectors will give the resultant as a vector. Cross product of two vectors is calculated by right hand rule. We can denote this using X. For avoiding confusion, we will use `@` also. Right hand rule is nothing but the resultant of any two vectors is perpendicular to the other two vectors. Using cross product, we can find the magnitude of the resultant vector. The usage of right hand rule is to find the direction of the resultant.


Properties of Cross Product

Cross product of two vectors is indicated like the following  `vecX xx vecY = |vecX|.|vecY| sin theta`

Let us take any two vectors `vecX = x veci + y vecj + z veck`    and `vecy = a veci + b vecj + c veck`

So cross product of these two vectors can be defined by matrices form. This is called determinant form.

`vecX xx vecY = [[veci, vecj,veck],[x,y,z],[a,b,c]]`

`vecX xx vecY = veci (yc - zb) - vecj (xc - za) + veck (xb - ya)`

This is the answer for the cross product of two vectors.

The area of the parallelogram is the cross product.

Properties of Cross product:

1. The Cross product is an anti commutative property

It mean `vecA xx vecB = - vecB xx vecA`

2. The cross product is having the distributive property over addition.

`vecA xx (vecB + vecC) = vecA xx vecB + vecA xx vecC`

3. Cross product doesn't have associative property

4. Cross product will satisfy the Jacobi property.

`vecA xx (vecB xx vecC) + vecB xx (vecC xx vecA) + vecC xx (vecA xx vecB) = 0 `

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Examples of Cross Product of Two Vectors


Example 1: Find the cross product of the following two vectors. `vecX = 5 veci + 6 vecj + 2 veck` And `vecY = veci +vecj +veck`

Solution: To find the cross product of two vectors we have to write the given vectors in determinant form. Using the determinant form we can find the Cross product of two vectors.

`vecX = 5 veci + 6 vecj + 2 veck` and `vecY = veci +vecj +veck`

`vecX xx vecY = [[veci,vecj,veck],[5, 6, 2],[1,1,1]]`

`= (6 - 2)veci - (5 - 2 ) vecj + (5 - 6) veck`

` = 4 veci - 3 vecj - veck`

Using right hand rule we can find the direction of the resultant vector. Cross product is used to calculate the force and displacement in physics. Using the right hand rule we can find the direction of the force and displacement. The cross product result is the area of the parallelogram. We can calculate the magnitude of the resultant vector by taking the square roots of the vectors.