Question

Two vectors A and B are defined as A=ai+bj and B=bi−aj​, where a is a constant. If the vector sum A+B has its maximum magnitude of ​a/√2, find the value of b.

a) b=a
b) b=−a
c) b=​a/√2​
d) b=a/−√2​​

Answer 1

d) b = a/-√2. This value of b makes the magnitude of A + B equal to the maximum of a/√2.

How to find the value?

a) b = a:

In this case, B = bi - ai = a(i - j). Adding A and B would result in (a + a)i + (b - a)j = 2ai + (b - a)j

This doesn't satisfy the condition of the maximum magnitude being a/√2.

b) b = -a:

Here, B = -bi - ai = -a(i + j). Adding A and B gives (a - a)i + (b + a)j = 2aj.

Again, this doesn't meet the maximum magnitude requirement.

c) b = a/√2:

With b = a/√2, B becomes bi - ai = a/√2 (i - j). Adding A and B results in (a + a/√2)i + (b - a/√2)j = (a√2 + a)i.

This still doesn't fulfill the maximum magnitude condition.

d) b = a/-√2:

Finally, when b = -a/√2, B becomes -bi + ai = a/√2 (i + j). Adding A and B leads to (a - a/√2)i + (b + a/√2)j = (a√2 - a)j.

This satisfies the condition as the magnitude of (a√2 - a)j is a/√2.

Therefore, the correct answer is d) b = a/-√2. This value of b makes the magnitude of A + B equal to the maximum of a/√2.