Question

How do you find the inner product and state whether the vectors are perpendicular given `<-5,3>*<2,-3>`?

Answer 1

Use the fact that the dot product of two vectors that are perpendicular is equal to zero.

Explanation:

The inner product (more commonly called the dot product) of two vectors of dimension n:

`vecA = < A_1, A_2, ..., A_n> and vecB = < B_1, B_2, ..., B_n>`

Is computed as follows:

`vecA*vecB = (A_1)(B_1) + (A_2)(B_2) +, ..., + (A_n)(B_n)`

The given vectors are of dimension 2:

Let `vecA = <-5, 3 >`
Let `vecB = <2, -3 >`

`vecA*vecB = (A_1)(B_1) + (A_2)(B_2)`

`vecA*vecB = (-5)(2) + (3)(-3)`

`vecA*vecB = -19`

Another way to compute the dot product is:

`vecA*vecB = |vecA||vecB|cos(theta)`

where:

`|vecA| = sqrt(A_1^2 + A_2^2 +, ..., + A_n^2)`,

`|vecB| = sqrt(B_1^2 + B_2^2 +, ..., + B_n^2)`,

and `theta` is the angle between the two vectors.

If all that you care about is whether the vector are perpendicular, then you do not care about the magnitudes; you only care whether (or not) the dot product is zero.

If the two vectors were perpendicular, then `theta = pi/2 (or 90^@)` and the `cos(theta) = 0`. This means that the dot product would be zero.

Because the dot product of the given vectors is NOT zero, one can say that the vectors are not perpendicular.