Question

Help.. Given
,
,
, and
. Determine the magnitude of
.

Answer 1

Given:

`||\vec u||=8, \ \theta_{\vec {u}}=55 \textdegree`

`||\vec v||=6, \ \theta_{\vec {v}}=40 \textdegree`

Find:

`||\vec u + \vec v || = \ ??`

In order to complete this problem we first have split each vector given in magnitude-angle form into its components.

`\vec u = < \vec u_x, \vec u_y > = < ||\vec u||cos \theta_{\vec {u}},||\vec u||sin \theta_{\vec {u}} >`

`\vec v = < \vec v_x, \vec v_y > = < ||\vec v||cos \theta_{\vec {v}},||\vec v||sin \theta_{\vec {v}} >`

For vector u:

`\vec u = < (8)cos(55\textdegree),(8)sin (55\textdegree) > \Longrightarrow \boxed{\vec u = < 4.589,6.553 > }`

For vector v:

`\vec v = < (6)cos(40\textdegree),(6)sin (40\textdegree) > \Longrightarrow \boxed{\vec v = < 4.596,3.857 > }`

Now we have vectors u and v split into their x and y components. We can now add these vectors.

`\vec u + \vec v = < \vec{u_x}+\vec{v_x},\vec{u_y}+\vec{v_y} >`

`\Longrightarrow \vec u + \vec v = < 4.589+4.596,6.553+3.857 > \Longrightarrow \boxed{\vec u + \vec v = < 9.185,10.41 > }`

The question asks for the magnitude of vectors u plus v. So,

`||\vec u + \vec v|| = √(((\vec u + \vec v)_x)^2+((\vec u + \vec v)_y)^2)` and the angle,
`\theta=tan^(-1)(((\vec u + \vec v)_y)/((\vec u + \vec v)_x) )`

`\Longrightarrow ||\vec u + \vec v|| = √((9.185)^2+(10.41)^2) \Longrightarrow \boxed`

`\Longrightarrow\theta=tan^(-1)((10.41)/(9.185) ) \Longrightarrow\boxed{\theta=48.577 \textdegree}`

Thus,
`\boxed{ ||\vec u + \vec v||=13.883 \ at \ {\theta=48.577 \textdegree}} \therefore Sol.`