Question

Suppose X and Y are vectors in R^n that have the same length. Show that X ⋅ Y bisects the angle between X and Y.

(A question about specific vector properties and the angle between vectors would need to be provided for options)
A) True
B) False
C) Depends on the specific vectors
D) Not enough information to determine

Answer 1

The dot product of vectors X and Y is proportional to cos(θ), where θ is the angle between X and Y. Therefore, X ⋅ Y bisects the angle between X and Y.

Step-by-step explanation:

In order to show that X ⋅ Y bisects the angle between X and Y, we need to show that the angle between X ⋅ Y and X is half the angle between X and Y. Let's denote θ as the angle between X and Y. The dot product of two vectors is defined as the product of their magnitudes and the cosine of the angle between them:

X ⋅ Y = |X||Y|cos(θ)

We also know that the lengths of vectors X and Y are the same, so |X| = |Y|. Substituting this into the equation gives:

X ⋅ Y = |X||X|cos(θ) = |X|^2cos(θ)

This shows that X ⋅ Y is proportional to cos(θ), which means that the angle between X ⋅ Y and X is equal to θ/2, proving that X ⋅ Y bisects the angle between X and Y.