problem: a) find q in vector form, along ad. b) find the component of q along the x-, y- and z - axis (qx…

problem: a) find q in vector form, along ad. b) find the component of q along the x-, y- and z - axis (qx, qy, qz) hint: solve in terms of distance a and vector magnitude q. for clarification: - rectangle obdc lies on the y - z plane - triangle oab lies on the x - y plane - triangle oac lies on the x - z plane

problem: a) find q in vector form, along ad. b) find the component of q along the x-, y- and z - axis (qx, qy, qz) hint: solve in terms of distance a and vector magnitude q. for clarification: - rectangle obdc lies on the y - z plane - triangle oab lies on the x - y plane - triangle oac lies on the x - z plane

Answer

Explanation:

Step1: Find unit - vector along AD

First, find the coordinates of points A, D. Let O be the origin (0,0,0). Point A has coordinates (2a,0,0), point D has coordinates (0,a,2a). The vector $\overrightarrow{AD}=\left(0 - 2a,a - 0,2a-0\right)=(- 2a,a,2a)$. The magnitude of $\overrightarrow{AD}$ is $|\overrightarrow{AD}|=\sqrt{(-2a)^2+a^2+(2a)^2}=\sqrt{4a^{2}+a^{2}+4a^{2}} = 3a$. The unit - vector $\hat{u}$ along $\overrightarrow{AD}$ is $\hat{u}=\frac{\overrightarrow{AD}}{|\overrightarrow{AD}|}=\left(-\frac{2}{3},\frac{1}{3},\frac{2}{3}\right)$.

Step2: Find vector Q in vector form

Since $\overrightarrow{Q}=Q\hat{u}$, then $\overrightarrow{Q}=Q\left(-\frac{2}{3}\hat{i}+\frac{1}{3}\hat{j}+\frac{2}{3}\hat{k}\right)=-\frac{2Q}{3}\hat{i}+\frac{Q}{3}\hat{j}+\frac{2Q}{3}\hat{k}$.

Step3: Find components of Q

The component of $\overrightarrow{Q}$ along the x - axis $Q_x=-\frac{2Q}{3}$, along the y - axis $Q_y=\frac{Q}{3}$, and along the z - axis $Q_z=\frac{2Q}{3}$.

Answer:

a) $\overrightarrow{Q}=-\frac{2Q}{3}\hat{i}+\frac{Q}{3}\hat{j}+\frac{2Q}{3}\hat{k}$ b) $Q_x = -\frac{2Q}{3},Q_y=\frac{Q}{3},Q_z=\frac{2Q}{3}$