Question

Using the component method of adding vectors, find the resultant of the following vectors: 50m to the right, 40m down, and 30m, 30o east of north.

Answer 1

Explanations & Calculations

• By horizontal resolution,

$\qquad\qquad \begin{aligned} \to \\ \small \vec{X}&=\small 30m\sin30+50m\\ &=\small 15m+50m\\ &=\small 65m \end{aligned}$

• By vertical resolution,

$\qquad\qquad \begin{aligned} \uparrow\\ \small \vec{Y}&=\small 30m\cos30-40m\\ &=\small 15\sqrt3m-40m\\ &=\small 5(3\sqrt3-8)m\\ &=\small -14.02m \end{aligned}$

• This negative sign implies that the Y resolution is not to the right, it is to the left.
• Now the resultant can be found by

$\qquad\qquad \begin{aligned} \small R^2&=\small X^2+Y^2\\ &=\small (65m)^2+(-14.02m)^2\\ &=\small 4421.56m^2\\ \small R&=\small \sqrt{4421.56m^2}\\ &=\small 66.49m\\ &\approx\small 66m \end{aligned}$

• The direction of the resultant is

$\qquad\qquad \begin{aligned} \small \theta &=\small \tan^{-1}\Big(\frac{Y}{X}\Big)\\ &=\small \tan^{-1}\Big(\frac{-14.02m}{65m}\Big)\\ &=\small -12.17^0 \end{aligned}$

• Then the direction is $\small 12.17^0$ North of west.