2. If A = 8i- 8j. B = -i +6j and C = -25. Find the constants a and b such that aA+ bB + C = 0.

Answer 1

To find the constants a and b such that aA + bB + C = 0, we can equate the components of the vectors.

Given:
A = 8i - 8j
B = -i + 6j
C = -25

Let's equate the components:

For the x-component:
a(8) + b(-1) + (-25) = 0
8a - b - 25 = 0

For the y-component:
a(-8) + b(6) + 0 = 0
-8a + 6b = 0

Now, we have a system of equations:

8a - b - 25 = 0 ---(1)
-8a + 6b = 0 ---(2)

To solve this system of equations, we can use the method of substitution.

From equation (2), we can express -8a as 6b:

-8a = -6b
a = (6/8)b
a = (3/4)b

Now, substitute this value of a into equation (1):

8(3/4)b - b - 25 = 0
6b - b - 25 = 0
5b - 25 = 0
5b = 25
b = 5

Substitute the value of b = 5 into equation (2) to find a:

-8a + 6(5) = 0
-8a + 30 = 0
-8a = -30
a = 30/8
a = 3.75

Therefore, the constants in the expression aA + bB + C = 0 are a = 3.75 and b = 5.