Question

Let $\overrightarrow{B}=\left(5.0m,{30}^{\circ }counterclockwisefromvertical\right).$Find the $xandy$-components of localid="1651138630094" $\overrightarrow{B}$in each of the two coordinate systems shown in FIGURE EX3.18.

Short answer

(a) $\overrightarrow{B}=\left(-2.5m\right)\hat{i}+\left(4.3m\right)\hat{j}.$

(b)$\overrightarrow{B}=\left(0m\right)\hat{i}+\left(5m\right)\hat{j}.$

Step-by-step solution

Part (a) Step 1. Given information

The given figure is,

and$\overrightarrow{B}=\left(5.0m,{30}^{\circ }counterclockwisefromvertical\right).$We need to determine the$x$and$y$components of$\overrightarrow{B}$in coordinate shown in the figure.

Step 2. Solution

The vector $\overrightarrow{B}=\left(5.0m,{30}^{\circ }counterclockwisefromvertical\right)$falls into the second quadrant with the magnitude of $5m.$

$\overrightarrow{B_x}=\overrightarrow{B}\sin \varphi .$

$B_x=5\times \sin {30}^{\circ }.$

$=5\times \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.=-2.5m.$

role="math" localid="1649274344760" $\overrightarrow{B_y}=\overrightarrow{B}\cos \varphi .$

$=5\times \cos {30}^{\circ }.$

$=5\times \raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right..$

$=\left(4.3m\right)\hat{j}.$

$\overrightarrow{B}=\left(-2.5m\right)\hat{i}+\left(4.3m\right)\hat{j}.$

Part (b) Step 1. Given information

The given figure is,

and$\overrightarrow{B}=\left(5.0m,{30}^{\circ }counterclockwisefromvertical\right).$We need to determine the$xandy$components of$\overrightarrow{B}$in the coordinate system.

Step 2. Solution

The vector $\overrightarrow{B}=\left(5.0m,{30}^{\circ }counterclockwisefromvertical\right).$The $xandy$axis are rotated ${30}^{\circ }.\overrightarrow{B}$perfectly lies in the $y$-axis.

$\varphi =0^{\circ }.$

$\overrightarrow{B_x}=\overrightarrow{B}\sin \varphi .$

$=5m\times \sin \left(0^{\circ }\right).$

$=\left(0m\right)\hat{i}.$

$\overrightarrow{B_y}=\overrightarrow{B}\cos \varphi .$

$=5\times 1.$

$=\left(5m\right)\hat{j}.$

$\overrightarrow{B}=\left(0m\right)\hat{i}+\left(5m\right)\hat{j}.$