Chapter 9.2, Problem 1E

To determine

To fill: The dot product of two nonzero vectors and the nature of the two vectors.

Answer to Problem 1E

The dot product of two nonzero vectors is $u\cdot v=a_1{b}_1+a_2{b}_2$ and the dot product of two vectors is scalar.

Explanation of Solution

The two nonzero vectors u is $\langle a_1,a_2\rangle$ and v is $\langle b_1,b_2\rangle$ and the angle between the vectors is $\theta$.

Multiply and add the corresponding components of vectors u and v to find the dot product of vectors.

$\begin{array}{c}u\cdot v=\langle a_1,a_2\rangle \cdot \langle b_1,b_2\rangle \\ =a_1{b}_1+a_2{b}_2\end{array}$

The dot product of two nonzero vectors is a real number or scalar.

Thus, the dot product of two nonzero vectors is $u\cdot v=a_1{b}_1+a_2{b}_2$ and the dot product of two vectors is scalar.