Expert: Sherry Wallin - 3/17/2009

Question
Can you please calculate the dot product of each of the following pairs of vectors, and also state what the result means about the angle between the vectors(and show the working out on how to you got the answer, so I can work it out as well):

(i)  (i + j ). (2i +3j)

(ii) (3i – j + 2k ). (3i + 3j - 3k)

Thanks a lot!

Answer
John~

The dot product is just a number and you can only multiple 'like terms', this means i's with i's, k's with k's and j's with j's. The dot product divided by the product of the magnitudes of the vectors is equal to the cosine of the angle between the vectors.

in a formula this is: cos theta = [a dot b]/|a||b|
so when the angle is pi/2 the cos pi/2 is 0 you will get the dot product to be zero.

Your problem: (i+j)(2i+3j) = 1(2) + 1(3) = 5
if a = i + j then |a| =  (1^2 + 1^2)^1/2 = (1 + 1)^1/2 = sqrt(2)
if b = 2i + 3j then |b| = (2^2 + 3^2)^1/2 = (4+9)^1/2 = sqrt(13)

cos theta = 5/[sqrt(26)
You can find theta by taking the arccosine of both sides about 11.3 degrees.

I hope this thoroughly explains the process for you.

Math Prof