Chapter 11, Problem 6PS

To determine

Represent the force exerted on each leg of the tripod as a vector.

Answer to Problem 6PS

  $P{Q}_1$ = $\boxed{\langle 0,-1,-4\rangle }$ , $P{Q}_2$ = $\boxed{\langle \frac{\sqrt{3}}{2},\frac{1}{2},-4\rangle }$ and $P{Q}_3$ = $\boxed{\langle -\frac{\sqrt{3}}{2},\frac{1}{2},-4\rangle }$

Explanation of Solution

Given information:

A television camera weighing $120$ pounds is supported by a tripod (see figure). Represent the force exerted on each leg of the tripod as a vector.

  

Calculation:

Component form of any vector $v$ is the representation of vector $v$ which is directed by line segment from point $p\left(p_1,p_2,p_3\right)toQ\left(q_1,q_2,q_3\right)$ component form of $v$ is evaluated by subtracting initial coordinates $\left(p_1,p_2,p_3\right)$ from final coordinates $\left(q_1,q_2,q_3\right)$ . Mathematically

  $\begin{array}{l}v=\langle v_1,v_2,v_3\rangle \\ =pQ\\ =\langle \left(q_1-p_1\right),\left(q_2-p_2\right),\left(q_3-p_3\right)\rangle \end{array}$

The magnitude of is

  $\Vert v\Vert =\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2+{\left(z_2-z_1\right)}^2}$

  

From the above figure camera center point as $P\left(0,0,4\right)$ and its leg ends are $Q_1\left(0,-1,0\right),$

  $Q_2\left(\frac{\sqrt{3}}{2},\frac{1}{2},0\right)and{Q}_3\left(-\frac{\sqrt{3}}{2},\frac{1}{2},0\right).$ the leg vectors are directed from center points towards end points. Hence, the component form of these vectors is as follows:

Vector $p{Q}_1$ is

  $\begin{array}{l}p{Q}_1=\langle \left(0-0\right),\left(-1-0\right),\left(0-4\right)\rangle \\ =\boxed{\langle 0,-1,-4\rangle }\end{array}$

Vector $p{Q}_2$ is

  $\begin{array}{l}p{Q}_2=\langle \left(\frac{\sqrt{3}}{2},\frac{1}{2}-0\right),\left(0-4\right)\rangle \\ =\boxed{\langle \frac{\sqrt{3}}{2},\frac{1}{2},-4\rangle }\end{array}$

Vector $p{Q}_{3}is$

  $\begin{array}{l}p{Q}_3=\langle \left(-\frac{\sqrt{3}}{2}-0\right),\left(\frac{1}{2}-0\right),\left(0-4\right)\rangle \\ =\boxed{\langle -\frac{\sqrt{3}}{2},\frac{1}{2},-4\rangle }\\ \\ \end{array}$