The angle between the two vectors A and B is θ.

Suppose the inequality |A+B|<|A−B| exists.

This implies the magnitude of A−B is greater than the magnitude of A+B.

It further implies that the vector B of higher magnitude when subtracted from the vector A of lower magnitude, the resultant vector with greater magnitude is obtained.

However, it’s not just the magnitude but the signs of the components of the vectors.

When the vector B of opposite sign to that of the vector A, is subtracted from A; it adds to the value of A, hence magnitude of the resultant vector increases

When the vector B of opposite sign to that of the vector A, is added to A; it reduces the value of A, hence the magnitude of the resultant vector decreases.

Due to change in signs of the components of two vectors, the angle between the two vectors is always greater than 90°.

Example: A=3ax+2ay and B=−ax−3ay

Both the vectors lie in different quadrant, hence the angle between them is greater than 90°.

A+B=3ax+2ay−ax−3ay=2ax−ay|A+B|=22+(−1)2=4+1=5

A−B=3ax+2ay−(−ax−3ay)=4ax+5ay|A−B|=42+52=16+25=41

Thus, the inequality |A+B|<|A−B| is verified.

Therefore, when |A+B|<|A−B|, the angle between the two vectors A and B  is θ>90°.

The angle between the two vectors A and B is θ.

Suppose the inequality |A+B|=|A−B| exists.

Example: A=ax and B=ay

Both the vectors are on the axis, hence the angle between them is 90°.

A+B=ax+ay|A+B|=12+12=1+1=2

A−B=ax−ay|A−B|=12+(−1)2=1+1=2

Thus, the inequality |A+B|=|A−B| is verified.

Therefore, when |A+B|=|A−B|, the angle between the two vectors A and B  is 90°.

The angle between the two vectors A and B is θ.

Suppose the inequality |A+B|>|A−B| exists.

This implies the magnitude of A+B is greater than the magnitude of A−B.

It further implies that the vector B when subtracted from the vector A, the resultant vector with greater magnitude is obtained.

However, it’s not just the magnitude but the signs of the components of the vectors.

When the vector B of same sign to that of the vector A, is added to A; it adds to the value of A, hence magnitude of the resultant vector increases

When the vector B of same sign to that of the vector A, is subtracted from A; it reduces the value of A, hence the magnitude of the resultant vector decreases.

As the sign of the components of two vectors are same, the angle between the two vectors is always less than 90°.

Example: A=−3ax+2ay and B=−ax+3ay

Both the vectors lie in same quadrant, hence the angle between them is less than 90°.

A+B=−3ax+2ay−ax+3ay=−4ax+5ay|A+B|=(−4)2+52=16+25=41

A−B=−3ax+2ay+ax−3ay=−2ax−ay|A−B|=(−2)2+(−1)2=4+1=5

Thus, the inequality |A+B|>|A−B| is verified.

Therefore, when |A+B|>|A−B|, the angle between the two vectors A and B  is θ<90°.