Problem 75 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Although I expected vector operations to produce another vector, the dot product of two vectors is not a vector, but a real number.

Step-by-Step Solution

Answer

The statement 'Although I expected vector operations to produce another vector, the dot product of two vectors is not a vector, but a real number.' makes sense because it accurately describes the properties of the dot product in vector operations.

Step 1: Definition of Dot Product

The dot product is a type of multiplication that takes two vectors and returns a scalar (or real number), not another vector. The dot product of vectors \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k} \) is given by \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \).

Step 2: Examination of Statement

The statement says 'the dot product of two vectors is not a vector, but a real number.' Comparing this with our definition from Step 1, we see that this statement is indeed correct. The dot product operation does result in a real number, not a vector.

Step 3: Final Analysis

From the illustrating the meaning of dot product and comparing it with the statement given, we conclude that the statement makes sense because it accurately describes the result of a dot product of two vectors.