Q1.4P - Introduction to Electrodynamics

Question

Use the cross product to find the components of the unit vector $\hat{\mathit{n}}$ perpendicular to the shaded plane in Fig. 1.11.

Step-by-Step Solution

Answer

The unit vector is $\hat{n}$.

Step 1: Explain the concept and draw the plane using given information

To find the vector perpendicular to a plane, the boundary vector must be comuted and then cross product of the boundary vector of the concerned plane must be evaluated. The given plane is drawn as follows:

The plane in y-z plane has unit vector $\hat{n}$ , perpendicular to it as shown in the figure.

Step 2: Assume the vectors

The end points of the base A of given plane are $\left(0,0,0\right)$ and $(1,0,0)$ and end points of the base B, which is left of given plane are $(0,0,3)$ and $(1,0,0)$,  

 

Find the position vector of base A and B.

$$\begin{gathered} A=(0-1)i+(2-0)j+(0-0)k \\ =-i+2j+0k \end{gathered}$$

Solve for vector B.

$$\begin{gathered} B=(0-1)i+(2-0)k+(0-0)k \\ =-i+0j+3k \end{gathered}$$

Step 3: Find the cross product between A and B .

The formula of the cross product of the vectors $A$ and $B$ is

$A\times B=\left|A\right|\left|B\right|\sin \theta \hat{n}$, $\theta$ is the angle between the vectos $A$ and $B$.

Find the cross product of the vectors $A$ and $B$.

$$\begin{gathered} A\times B=\left|\begin{array}{ccc}i& j& k\\ -1& 2& 0\\ -1& 0& 3\end{array}\right| \\ =i(6-0)-j(-3-0)+k(0+2) \\ =6i+3y+2k \end{gathered}$$

The unit vector $\hat{n}$ is obtained as

$$\begin{gathered} \hat{n}=\frac{A\times \mathbf{\hspace{0.17em}}B}{\left|A\times B\right|} \\ =\frac{6i+3j+2k}{\sqrt{6^2+3^2+2^2}} \\ =\frac{6i+3j+2k}{\sqrt{49}} \\ =\frac{6i+3j+2k}{7} \\ =\frac{6}{7}i+\frac{3}{7}j+\frac{2}{7}k \end{gathered}$$

Thus, the unit vector is $\hat{n}=\frac{6}{7}i+\frac{3}{7}j+\frac{2}{7}k$.