Question

Clayton State University Department of Chemistry and Physics January 23, 2019 Physics 3454-Quiz 2 Name Pho T 1. Express the force as a Cartesian vect F 900 N 2 m 7 m 2. Project this force onto an OA direction where O is the origin of the coordinate system.

Answer 1

1. The force vector can be calculated by finding the magnitude and the direction of the vector. The magnitude of the force is given as 900N and the force originates at the point A (2,0,4) and ends at point B (-2,7,0). Therefore the unit vector along that direction is,

$\widehat{r} = \frac{(-2-2)\widehat{i} + 7 \widehat{j} -4 \widehat{k}}{\sqrt{(-2-2)^2 + 7^2 + (-4)^2)}} = -0.45 \widehat{i} + 0.78 \widehat{j} - 0.45 \widehat{k}$

Therefore the force vector is,

$\overrightarrow{F} = \left | \overrightarrow{F} \right | \widehat{r} = 900( -0.45 \widehat{i} + 0.78 \widehat{j} - 0.45 \widehat{k}) = -405 \widehat{i} + 702 \widehat{j} - 405 \widehat{k}$

2. For projection, we have to find the vector OA, which can be calculated by using the coordinate points of O and A, and then take dot product with the F vector to get the projection.

The vector OA is, (We know the coordinate A is (2,0,4)).

$\overrightarrow{OA} = 2 \widehat{i} + 4 \widehat{k}$

Now taking the dot product we get, the projection as,

$\Rightarrow \overrightarrow{OA} \cdot \overrightarrow{F} = (2 \widehat{i} + 4 \widehat{k}) \cdot (-405 \widehat{i} + 702 \widehat{j} - 405 \widehat{k}) = -2430$