Problem 76 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I've noticed that whenever the dot product is negative, the angle between the two vectors is obtuse.

Step-by-Step Solution

Answer

Yes, the statement makes sense because the dot product being negative implies that the angle between the vectors is obtuse (greater than 90° and less than 180°).

Step 1: Understanding Dot Product

In a Euclidean space, the dot product of two vectors \( \vec{A} \) and \( \vec{B} \) is defined as \( \vec{A} . \vec{B} = ||\vec{A}|| ||\vec{B}|| cos\theta \), where ||\vec{A}|| and ||\vec{B}|| are the magnitudes of \( \vec{A} \) and \( \vec{B} \) respectively, and \( \theta \) is the angle between them.

Step 2: Determining the Sign of the Dot Product

The cosine function is positive for angles less than 90° and negative for angles greater than 90°. Therefore, if the dot product is negative, the cosine of the angle between the two vectors must be negative.

Step 3: Concluding the Statement

Since cos(\( \theta \)) is negative when 90° < \( \theta \) < 180°, a negative dot product indeed implies that the angle between two vectors is obtuse. Therefore, the statement 'Whenever the dot product is negative, the angle between the two vectors is obtuse' does make sense.