Question

How do you find the cross product and state whether the resulting vectors are perpendicular to the given vectors `<-1,-3,2>times<6,-1,-2>`?

Answer 1

The cross product of two vectors will always result in a third vector that is perpendicular to both. You can check that is it perpendicular by verifying that the dot-product with either vector is zero.

Explanation:

I use a 5 column determinant to compute the cross-product:

#| (hati, hatj, hatk, hati, hatj), (-1,-3,2,-1,-3), (6,-1,-2,6,-1) | = #

Add the product of the diagonal descending to the right and subtract the product of the diagonal descending to the left.

`{-3(-2) - (2)(-1)}hati + {2(6) - (-1)(-2)}hatj + {-1(-1) - (-3)(6)}hatk =`

Simplify:

`8hati + 10hatj + 19hatk`

Convert the vector notation: `<8,10,19>`

Verify that it is perpendicular to the first vector:

`<8,10,19>•<-1, -3, 2> = 8(-1) + 10(-3) + 19(2) = 0`

The dot-product is 0, therefore, the two vectors are perpendicular.

Verify that it is perpendicular to the second vector:

`<8,10,19>•<6, -1, -2> = 8(6) + 10(-1) + 19(-2) = 0`

The dot-product is 0, therefore, the two vectors are perpendicular.