Question

How do you find the inner product and state whether the vectors are perpendicular given `<7,-2,4>*<3,8,1>`?

Answer 1

Dot product = 9. Vectors are not orthogonal.

Explanation:

`vec(u)*vec(v) = sum_(i=1)^n u_iv_i " for " vec(u),vec(v) in RR^n`

So, in `RR^3` the dot product will be given by `sum_(i=1)^3u_iv_i`

`=7*3 + (-2)*8 + 4*1 = 9`

For orthogonality, think about the other definition of the dot product:

`vec(u)*vec(v) = |vec(u)||vec(v)|costheta`

If the vectors are orthogonal, `theta = pi/2` and the dot product is zero. This is how you can check orthogonality with the dot product. As you can see, the dot product of these two vectors is non-zero and thus they are not orthogonal.