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

1 Answer
Jan 28, 2016

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]#