Question

Suppose you add two vectors A and B . What relative direction between them produces the resultant with the greatest magnitude? What is the maximum magnitude? What relative direction between them produces the resultant with the smallest magnitude? What is the minimum magnitude?

Answer 1

Answer:

Same direction to produce maximum magnitude and opposite direction to produce minimum magnitude

Explanation:

Let a be the angle between vectors A and B. Generally when we add A to B, we can split A into 2 sub vectors, 1 parallel to B and the other perpendicular to B.

Also let A and B be the magnitude of vector A and B, respectively.

We have the parallel component after addition be

Acos(a) + B

And the perpendicular component after addition be

Asin(a)

The magnitude of the resulting vector would be

$\sqrt{(Acos(a) + B)^2 + (Asin(a))^2}$

$= \sqrt{A^2cos^2a + B^2 + 2ABcos(a) + A^2sin^2a}$

$= \sqrt{A^2(cos^2a + sin^2a) + B^2 + 2ABcos(a)}$

$= \sqrt{A^2 + B^2 + 2ABcos(a)}$

As A and B are fixed, the equation above is maximum when cos(a) = 1, meaning a = 0 degree and vector A and B are in the same direction, and minimum with cos(a) = -1, meaning a = 180 degree and vector A and B are in opposite direction.