Question

Please help. Need help with 2d and vector problems. Please help it's urgent?

Answer 1

Let
`\theta` denote the angle made by a given force with the positive x-axis (pointing right). So in (1), A makes an angle of 90º - 30º = 60º, and B makes an angle of -45º; and in (2), A still makes angle of 30º, B makes an angle of -15º, and C makes one of -90º.

Separate each force vector in to x- and y-components, then compute the required vector. This involves writing each given vector
`\mathbf v` as

`\mathbf v=v_x\,\mathbf i+v_y\,\mathbf j`

where

`\begin{cases}v_x=\|\mathbf v\|\cos\theta\\v_y=\|\mathbf v\|\sin\theta\end{cases}`

(1)

`\mathbf A=200\left(\cos60^\circ\,\mathbf i+\sin60^\circ\,\mathbf j\right)=100\,\mathbf i+100\sqrt3\,\mathbf j`

`\mathbf B=150\left(\cos(-45^\circ)\,\mathbf i+\sin(-45^\circ)\,\mathbf j\right)=75\sqrt2\,\mathbf i-75\sqrt2\,\mathbf j`

`\implies\mathbf A-\mathbf B\approx-6.066\,\mathbf i+67.139\,\mathbf j`

(2)

`\mathbf A=25\sqrt3\,\mathbf i+25\,\mathbf j`

`\mathbf B=25\sqrt2(1+\sqrt3)\,\mathbf i+25\sqrt2(1-\sqrt 3)\,\mathbf j`

`\mathbf C=-50\,\mathbf j`

`\implies\mathbf A+\mathbf B+\mathbf C\approx139.894\,\mathbf i-50.882\,\mathbf j`

`\implies\|\mathbf A+\mathbf B+\mathbf C\|\approx148.86`