# Aidan Sharpe - Homework 1

1. $\vec{F}$ has magnitude $800$N and makes an angle of $35^\circ$ with the y-axis in the second quadrant.
    
    $\vec{F}_x = \lVert \vec{F} \rVert \sin(-35^\circ) = 800 \times -0.5736 = -458.861$

    $\vec{F}_y = \lVert \vec{F} \rVert \cos(-35^\circ) = 800 \times 0.8192 = 655.322$

    $\vec{F} = -458.861\hat{i} + 655.322\hat{j}$

1. The force, $\vec{F}$, has magnitude $6.6$kN and slope is $-\frac{5}{12}$.
    
    $\theta = \arctan \left(-\frac{5}{12} \right) = -0.3948 = -22.620^\circ$

    $\vec{F}_x = 6600\cos(-22.620^\circ) = 6600 \times 0.9231 = 6092.308$

    $\vec{F}_y = 6600\sin(-22.620^\circ) = 6600 \times -0.3846 = -2538.462$

    $\vec{F} = 6092.308\hat{i} - 2538.462\hat{j}$

1. $F_1 = 500$N and $F_2 = 350$-N. $F_1$ is in the direction of the x-axis, and $F_2$ makes an angle $60^\circ$ with the x-axis.

    $\vec{R} = \vec{F}_1 + \vec{F}_2$

    $\vec{F}_1 = 500\hat{i} + 0\hat{j}$
    
    $\vec{F}_2 = 350\cos(60^\circ)\hat{i} + 350\sin(60^\circ)\hat{j} = 175\hat{i} + 303.109\hat{j}$

    $\vec{R} = 675\hat{i} + 303.109\hat{j}$

    $\theta_R = \arctan(\frac{303.109}{675}) = 0.4221 = 24.182^\circ$

1. $F_y = 70$lbs. The slope of $\vec{F}$ is $\frac{12}{5}$.
    
    $\frac{70}{F_x} = \frac{12}{5}$

    $70 \times 5 = 12 F_x$

    $F_x = \frac{70 \times 5}{12} = 29.166$

    $\vec{F} = F_x\hat{i} + F_y\hat{j}$

    $\lVert \vec{F} \rVert = \sqrt{F_x^2 + F_y^2} = \sqrt{29.166^2 + 70^2} = 75.833$N

1. $\vec{F}_1$ has magnitude $2$kN and makes an angle $30^\circ$ with the x-axis. $\vec{F}_2$ has magnitude $3$kN and has slope $-\frac{4}{3}$.

    $\vec{F}_1 = 2000\cos(30^\circ)\hat{i} + 2000\sin(30^\circ)\hat{j} = 1732.051\hat{i} + 1000\hat{j}$

    $\theta_{F_2} = \arctan(-\frac{4}{3}) = -0.9273 = -53.130^\circ$

    $\vec{F}_2 = 3000\cos(\theta_{F_2})\hat{i} + 3000\sin(\theta_{F_2})\hat{j} = 1800\hat{i} - 2400\hat{j}$

    $\vec{R} = \vec{F}_1 + \vec{F}_2 = (1732.051 + 1800)\hat{i} + (1000 - 2400)\hat{j} = 3532.051\hat{i} - 1400\hat{j}$

    $\theta = \arctan(-\frac{1400}{3532.051}) = -0.3774 = -21.622^\circ$

    $\lVert \vec{R} \rVert = \sqrt{3532.051^2 + (-1400)^2} = 3799.393$