Question

If aƒ— and bƒ— are nonzero vectors for which aƒ— ⋅ bƒ—=0, it must follow that

1) aƒ— and bƒ— are parallel
2) aƒ— and bƒ— are perpendicular
3) aƒ— and bƒ— are collinear
4) aƒ— and bƒ— are linearly independent

Answer 1

Two non-zero vectors with a dot product of zero are perpendicular to each other. The correct answer to the question is that vectors a and b are perpendicular.

Step-by-step explanation:

If two non-zero vectors a and b have a dot product a · b = 0, it implies that the vectors are perpendicular, or orthogonal, to each other. The dot product is a measure of how much one vector extends in the direction of another, and when it is zero, it indicates that there is no extension of one vector in the direction of the other, hence they are at a 90-degree angle to each other.

The question, therefore, has one correct answer: 2) a and b are perpendicular.

It is important to note that being perpendicular is different from being parallel or collinear. Parallel vectors are in the same or opposite directions with a non-zero dot product if they are not zero vectors themselves. Collinear vectors lie on the same line, which means they could be parallel or anti-parallel. Lastly, linear independence refers to vectors that cannot be expressed as a scalar multiple of one another, which does not necessarily relate to their dot product being zero.