Question

Two vectors, $\overrightarrow{\mathbf{r}}$andlocalid="1656309851434" $\overrightarrow{\mathbf{s}}$, lie in the xy plane.Their magnitudes are 4.50 and 7.50units, respectively, and their directions are $\mathbf{320}\mathbf{°}$and $\mathbf{85}\mathbf{.}\mathbf{0}\mathbf{°}$, respectively, as measured counterclockwise from the positive x axis.What are the values of (a)$\overrightarrow{\mathbf{r}}\mathbf{.}\overrightarrow{\mathbf{s}}$, and (b)$\overrightarrow{\mathbf{r}}\mathbf{\times }\overrightarrow{\mathbf{s}}$?

Short answer

(a) The dot product ofvectors$\overrightarrow{r}$ and $\overrightarrow{s}$is$-18.8\mathrm{units}$.

(b) The cross product of vectors $\overrightarrow{r}$and $\overrightarrow{s}$is 26.9 units along the positive z-axis.

Step-by-step solution

Vector operations

Vector operations can be used to find the dot product and cross product between two vectors. The dot product of two vectors produces a scalar quantity whereas the cross product of two vectors results in a vector quantity.The cross product will be in the perpendicular direction to both the vectors and its direction can be found by using the right-hand rule.

The equations for the dot product and cross product are as below:

$\overrightarrow{a}·\overrightarrow{b}=\left|a\right|\left|b\right|\cos \theta \left(\mathrm{where}\mathrm{\theta }\mathrm{is}\mathrm{angle}\mathrm{between}\mathrm{them}\right)$ (i)

$\overrightarrow{a}\times \overrightarrow{b}=\left|a\right|\left|b\right|\sin \theta \left(\mathrm{where}\mathrm{\theta }\mathrm{is}\mathrm{angle}\mathrm{between}\mathrm{them}\right)$ (ii)

Here are the given quantities in the problem.

The magnitude of vector $\overrightarrow{r}$and $\overrightarrow{s}$, is 4.50, 7.30 respectively.

The angle of $\overrightarrow{r}$is $320°$and vector$\overrightarrow{s}$is$85°$counterclockwise from the positive x- axis.

(a) Finding the dot product of r→and s→.

Find the angle between $\overrightarrow{r}$and $\overrightarrow{s}$.

$$\begin{gathered} ∆\theta =\left(320°-85°\right) \\ =235° \end{gathered}$$

Now, substitute the values of the magnitude of vectors and angle in equation (i)

$$\begin{gathered} \overrightarrow{r}·\overrightarrow{s}=\left(4.50\times 7.30\right)\cos \left(235°\right) \\ =-18.8\mathrm{units} \end{gathered}$$

Thus, the dot product of the vectors $\overrightarrow{r}$and $\overrightarrow{s}$is$-18.8\mathrm{units}$ .

(b) Finding the cross product of r→ and s→.

The angle between $\overrightarrow{r}$and $\overrightarrow{s}$is $125°$if measured in counterclockwise direction from vector $\overrightarrow{r}$to vector $\overrightarrow{s}$.

Substitute the values in equation (ii) to calculate the cross product.

$$\begin{gathered} \overrightarrow{r}\times \overrightarrow{s}=\left|4.50\right|\times \left|7.30\right|\sin \left(125°\right) \\ =26.90\mathrm{units} \end{gathered}$$

Using the right-hand rule, it can be concluded that the direction of the cross product is along positive z-axis.

Thus, the cross product of$\overrightarrow{r}$ and $\overrightarrow{s}$is equal to 26.90 units in positive z-direction.