Question

What are the magnitude and direction of u+v+w ?

Answer 1

The magnitude and the direction of the resultant are approximately 57.871 and 198.676°.

How to determine the magnitude and the direction of a resultant

Vectors are elements with two given characteristics: Magnitude and direction. A resultant is derived from the sum of vectors, the magnitude is the norm of a vector and the direction is the orientation of a vector. The resultant and its characteristics are described below:

Resultant

$\vec r = \left(\sum\limits_{i=1}^{n}x_{i}, \sum\limits_{i=1}^{n}y_{i}\right)$     (1)

Magnitude

$\|\vec r\| = \sqrt{\left(\sum\limits_{i=1}^{n} x_{i}\right)^{2}+\left(\sum\limits_{i=1}^{n} y_{i}\right)^{2}}$   (2)

Direction

$\theta = \tan^{-1}\frac{\sum\limits_{i=1}^{n}y_{i}}{\sum\limits_{i=1}^{n}x_{i}}$   (3)

Where $\theta$ is resultant angle, measured in degrees.

If we know that $\|\vec u\| = 90$, $\theta_{u} = 40^{\circ}$, $\|\vec v\| = 60$, $\theta_{v} = 20^{\circ}$, $\|\vec w\| = 40$ and $\theta_{w} = 30^{\circ}$, then the resultant, its magnitude and its direction:

Resultant

$\vec r = (-90\cdot \cos 40^{\circ}-60\cdot \sin 20^{\circ}+40\cdot \cos 30^{\circ}, 40\cdot \sin 30^{\circ}+60\cdot \cos 20^{\circ}-90\cdot \sin 40^{\circ})$

$\vec r = (-54.824, 18.531)$

Magnitude

$\|\vec r\| = \sqrt{(-54.824)^{2}+18.531^{2}}$

$\|\vec r\| \approx 57.871$

Direction

$\theta = \tan^{-1} \left(\frac{18.531}{-54.824}\right)$

$\theta \approx 198.676^{\circ}$

The magnitude and the direction of the resultant are approximately 57.871 and 198.676°. $\blacksquare$

To learn more on vectors, we kindly invite to check this verified question: brainly.com/question/13322477