Question

Find the "conditions" for which the vector a+b bisects the angle between the non collinear vectors a and b

Answer 1

The conditions for which the vector a+b bisects the angle between the vector a+b bisects the angle between the non collinear vectors and b will be that a and b are mutually perpendicular to each other.

Thus is the geometrical proof in which a+b will be bisecting the vectors a and b if they are mutually perpendicular to each other.

Answer 2

The condition for which the vector a+b bisects the angle between the non collinear vectors a and b is that a and b has the same magnitude.

Imagine a rectangle. It has unequal sides x and y.

Now if we consider two vectors a and b along x and y respectively then one of its diagonal represents the sum vector a + b.

Consider a rectangle ABCD, sides x $\neq$ y.

Imagine ∠BAD and ∠DAC

• tan(∠BAD) = y/x

But

• tan(∠DAC) = x/y

This implies, ∠BAD $\neq$ ∠DAC

But if y = x , that means for a square

then ∠BAD = ∠DAC.

Therefore the vector a + b to bisect the angle between a and b ,  a and b should have equal magnitude.