Question

If two vectors have precisely equal magnitudes, their dot product is:

a) Always zero.

b) Always positive.

c) Always negative.

d) Either positive or negative.

Answer 1

The dot product of two vectors with equal magnitudes can be either positive or negative, depending on the angle between them. It is positive if the vectors are parallel and negative if they are antiparallel.

Step-by-step explanation:

If two vectors have precisely equal magnitudes, their dot product can be either positive or negative. The dot product is the result of the scalar multiplication of two vectors, also referred to as the scalar product. Specifically, the dot product is obtained by multiplying their magnitudes with the cosine of the angle between them.

If the vectors are parallel, the angle between them is 0 degrees, leading to a positive dot product as the cosine of 0 degrees is 1. If they are antiparallel, meaning they have a difference in direction of 180 degrees, the dot product is negative because the cosine of 180 degrees is -1.

For vectors that are at any other angle, the dot product will depend on that angle's cosine value. As such, the correct answer to the question is (d) Either positive or negative.