You are given vectors A = 5.0 ˆi - 6.5 ˆj and B = - 3.5 ˆi + 7.0 ˆj. A third vector C lies in the xy-plane. Vector C is perpendicular to vector A, and the scalar (dot) product of C with B is 15.0. From this information, find the components of vector C.

Answer: 4.29 and - 3.23

Explanation: Let vector C be represented as

C = ai + bj

First sentence is that vector A is perpendicular to vector C.

When 2 vectors are perpendicular, their dot product is zero, that is

A. C = 0

By doing so, we have that

A = 5i - 6.5j and C = ai + bj

A. C = (5i - 6.5j) * (ai + bj) = 0

5a (i. i) + 5b (i. j) - 6.5a (j. i) - 6.5b (j. j)

From vector dot product, i. i = j. j = 1 and i. j = j. i = 0

Hence we have that

A. C = 5a - 6.5b=0 ... equation 1

The second sentence is that the dot product of C and B is 15.0

C. B = (ai + bj) * (-3.5i + 7j) = 15

C. B = - 3.5a (i. i) + 7a (i. j) - 3.5b (i. j) + 7b (j. j)

But i. i = j. j = 1 and i. j = j. i = 0

C. B = - 3.5a + 7b = 15

-3.5a + 7b = 15 ... equation 2

5a - 6.5b=0

-3.5a + 7b = 15

From the first equation, we make 'a' the subject of formulae and we have that

5a = 6.5b

a = 6.5b/5 ... equation 3

Let us substitute the equation above into equation 2, we have that

-3.5 (6.5b/5) + 7b = 15

-22.75b/5 + 7b = 15

-4.55b = 15

b = 15 / - 4.55 = - 3.23.

Substitute b = - 3.23 into equation 3 to get the value for a

a = 6.5b/5

a = 6.5 (-3.23) / 5

a = 21.43/5 = 4.29

So the x component of (a) of vector C is 4.29 and the y component (b) is - 3.23