91Ó°ÊÓ

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:

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

$\stackrel{→}{a}×\stackrel{→}{b}=\left|a\right|\left|b\right|\sin \mathrm{θ}\left(\mathrm{where}\mathrm{θ}\mathrm{is}\mathrm{angle}\mathrm{between}\mathrm{them}\right)$ (ii)

Here are the given quantities in the problem.

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

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

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

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

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

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

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

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

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

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

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$\stackrel{→}{r}$ and $\stackrel{→}{s}$is equal to 26.90 units in positive z-direction.