Question

If a and bare two vectors such that 3i+2j +9k and b = i + aj + 3k. Find the value of a so that a+b is perpendicular to a-b.​

Answer 1

The value of a that makes vectors a+b and a-b perpendicular is ± 4√5.

Step-by-step explanation:

To find the value of a so that the vector a+b is perpendicular to the vector a-b, we use the fact that the dot product of two perpendicular vectors is zero. Given that a = 3i+2j+9k and b = i + aj + 3k, the vectors a+b and a-b are:

• a+b = (3+1)i + (2+a)j + (9+3)k = 4i + (2+a)j + 12k
• a-b = (3-1)i + (2-a)j + (9-3)k = 2i + (2-a)j + 6k

The dot product (a+b) · (a-b) is:

4·2 + (2+a)·(2-a) + 12·6 = 0

This simplifies to:

8 + (4 - a2) + 72 = 0

The above equation can be solved for a:

80 - a2 = 0

a2 = 80

a = ± √80 = ± 4√5

Therefore, the value of a that makes a+b and a-b perpendicular is a = ± 4√5.