Question

Evaluate the dot product of the three pairs of vectors in Figure

Short answer

$$\begin{gathered} \overrightarrow{A}·\overrightarrow{B}=11.5 \\ \overrightarrow{C}·\overrightarrow{D}=-4.6 \\ \overrightarrow{E}·\overrightarrow{F}=0 \end{gathered}$$

Step-by-step solution

Step 1. Given Information

We have

The dot product is the sum of the product of the component where this dot product is called the scalar product.

For two vector A and B and there is an angle x between then the dot product these two vectors is given by

$$\begin{gathered} \overrightarrow{A}·\overrightarrow{B}=AB\cos \alpha \\ \overrightarrow{C}·\overrightarrow{D}=CD\cos \alpha \\ \overrightarrow{E}·\overrightarrow{F}=EF\cos \alpha \end{gathered}$$

Step 2. Part a). A and B are the magnitudes and α is the angle

We are given vectors by $A=3$and $B=5$and the angle is role="math" localid="1648018330260" $\alpha =40°$

So,

$$\begin{gathered} \overrightarrow{A}·\overrightarrow{B}=AB\cos \alpha \\ =\left(3\right)\left(5\right)\cos 40° \\ =11.5 \end{gathered}$$

Step 3. Part b). C and D are the magnitudes and α is the angle 

The magnitude of the vector $C=2$and $D=3$and the angle is $\alpha =140°$

So,

$$\begin{gathered} \overrightarrow{C}·\overrightarrow{D}=CD\cos \alpha \\ =\left(2\right)\left(3\right)\cos 140° \\ =-4.6 \end{gathered}$$

Step 4. Part c). E and F are the magnitudes and α is the angle

The magnitude of the vector $E=3$and $F=4$and the angle is $\alpha =90°$

$$\begin{gathered} \overrightarrow{E}·\overrightarrow{F}=EF\cos \alpha \\ =\left(3\right)\left(4\right)\cos 90° \\ =0 \end{gathered}$$