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
