Question

What is the cross product of `<2 , 5 ,-3 >` and `<3 ,5 ,-2 >`?

Answer 1

To find the cross product of two vectors, `[a_1, a_2, a_3]` and `[b_1, b_2, b_3]`, which we will call the new vector `[c_1, c_2, c_3]`, we multiply and subtract as follows (careful, it's a little tricky!):

To give the first element of the cross product, ignore the first elements of each of the vectors being multiplied and multiply and subtract the remaining elements as follows:

`c_1 = a_2*b_3 - a_3*b_2`

The second element is the trickiest. Ignore the second elements of each vector, and multiply and subtract as shown:

`c_2=a_3*b_1-a_1*b_3`

Carefully note the order.

Finally, to find the third element of the resultant vector, ignore the third element of the two component vectors, and multiply and subtract as shown:

`c_3=a_1*b_2-a_2*b_1`

Putting it all together:

`[a_1, a_2, a_3] xx [b_1, b_2, b_3]`
= `[(a_2*b_3 - a_3*b_2)+(a_3*b_1-a_1*b_3)+(a_1*b_2-a_2*b_1)]`
= `[c_1, c_2, c_3]`

In the specific case we were asked about in the question:

`[2, 5, −3] xx [3, 5, −2]`

`= [(5*(-2) - (-3)*5)+((-3)*3-2*(-2))+(2*5-5*3)]`

= `[((-10) - (-15))+((-9)-(-4))+(10-15)]`

= `[5, -5, -5]`