# Step-by-Step Solution: Vector Addition by Tip-to-Tail Method Let's break down the problem and solve for the resultant vector: ## 1. **List the Given Vectors (with Directions)** - **$\vec{a}$**: 10 m, up - **$\vec{b}$**: 8 m @ 20° below the left horizon (i.e., 20° below $180^\circ$ from +x) - **$\vec{c}$**: 2 m, left - **$\vec{d}$**: 12 m @ 30° left of the vertical (i.e., 30° left from +y, or $120^\circ$ from +x) - **$\vec{f}$**: 8 N, right (assuming units are compatible, otherwise keep as 8 units right) - **$\vec{h}$**: 65 h, left (assuming units are compatible) - **$\vec{g}$**: 40 N @ 15° above the left horizon (not needed for this question) ## 2. **Required Vector Addition** \[ \vec{R} = 2\vec{a} - 2\vec{b} + \vec{h} - \vec{c} + \vec{f} - \vec{d} \] ## 3. **Convert All Vectors to Components** Assume all units are meters unless otherwise noted. Let’s assign directions: - Right: $+x$ - Up: $+y$ - Left: $-x$ - Down: $-y$ ### (a) $\vec{a}$: 10 m, up \[ \vec{a} = (, +10) \] So $2\vec{a} = (, +20)$ ### (b) $\vec{b}$: 8 m @ 20° below left horizon - Left horizon = $180^\circ$ - $20^\circ$ below = $180^\circ + 20^\circ = 200^\circ$ from +x \[ \vec{b}_x = 8\cos(200^\circ),\quad \vec{b}_y = 8\sin(200^\circ) \] \[ \cos(200^\circ) = -.9397,\quad \sin(200^\circ) = -.342 \] \[ \vec{b}_x = 8 \times (-.9397) = -7.5176 \] \[ \vec{b}_y = 8 \times (-.342) = -2.736 \] So $\vec{b} = (-7.52, -2.74)$ (rounded to two decimals) So $-2\vec{b} = -2 \times (-7.52, -2.74) = (15.04, 5.48)$ ### (h) $\vec{h}$: 65 h, left \[ \vec{h} = (-65, ) \] ### (c) $\vec{c}$: 2 m, left \[ \vec{c} = (-2, ) \] So $-\vec{c} = (2, )$ ### (f) $\vec{f}$: 8 N, right \[ \vec{f} = (8, ) \] ### (d) $\vec{d}$: 12 m @ 30° left of vertical - Vertical = $+y$ - 30° left = $90^\circ + 30^\circ = 120^\circ$ from +x \[ \vec{d}_x = 12\cos(120^\circ),\quad \vec{d}_y = 12\sin(120^\circ) \] \[ \cos(120^\circ) = -.5,\quad \sin(120^\circ) = .866 \] \[ \vec{d}_x = 12 \times (-.5) = -6 \] \[ \vec{d}_y = 12 \times .866 = 10.392 \] So $\vec{d} = (-6, 10.39)$ So $-\vec{d} = (6, -10.39)$ --- ## 4. **Sum All Components** \[ \vec{R} = 2\vec{a} - 2\vec{b} + \vec{h} - \vec{c} + \vec{f} - \vec{d} \] Sum x-components: - $2\vec{a}_x = $ - $-2\vec{b}_x = 15.04$ - $\vec{h}_x = -65$ - $-\vec{c}_x = 2$ - $\vec{f}_x = 8$ - $-\vec{d}_x = 6$ Total $x$: \[ R_x = + 15.04 - 65 + 2 + 8 + 6 = (15.04 + 2 + 8 + 6) - 65 = 31.04 - 65 = -33.96 \] Sum y-components: - $2\vec{a}_y = 20$ - $-2\vec{b}_y = 5.48$ - $\vec{h}_y = $ - $-\vec{c}_y = $ - $\vec{f}_y = $ - $-\vec{d}_y = -10.39$ Total $y$: \[ R_y = 20 + 5.48 + + + - 10.39 = 25.48 - 10.39 = 15.09 \] --- ## 5. **Resultant Magnitude and Direction** \[ |\vec{R}| = \sqrt{(-33.96)^2 + (15.09)^2} \] \[ |\vec{R}| = \sqrt{1154.24 + 227.73} = \sqrt{1381.97} \approx 37.2 \] **Direction $\theta$ (from +x axis, counterclockwise):** \[ \tan\theta = \frac{R_y}{R_x} = \frac{15.09}{-33.96} = -.4445 \] $\arctan(-.4445) \approx -23.9^\circ$ (since $R_x$ is negative, this is in the 2nd quadrant) So, \[ \theta = 180^\circ - 23.9^\circ = 156.1^\circ \] --- ## 6. **Scale for Graphing** - **Scale:** 1 cm = 2 units (meters) - **Length to draw:** $|\vec{R}| = 37.2\,\text{m} = 18.6\,\text{cm}$ --- ## 7. **Summary Table** | Vector | x-component | y-component | |---------------------|----------------|--------------| | $2\vec{a}$ | | 20 | | $-2\vec{b}$ | 15.04 | 5.48 | | $\vec{h}$ | -65 | | | $-\vec{c}$ | 2 | | | $\vec{f}$ | 8 | | | $-\vec{d}$ | 6 | -10.39 | | **Total** | **-33.96** | **15.09** | --- ## 8. **Final Answer** **Resultant Vector:** \[ \boxed{ |\vec{R}| \approx 37.2\,\text{units},\ \theta \approx 156^\circ\ \text{(from +x axis, CCW)} } \] - **To plot:** From origin, measure 18.6 cm at an angle of $156^\circ$ from the positive x-axis. - **Direction:** $156^\circ$ counterclockwise from the +x axis (or $24^\circ$ above the negative x-axis). --- ## 9. **Tip-to-Tail Graphic Process** - Draw each vector in sequence according to their components and angles, using the scale. - Start from origin, place each tip to the previous tail, following the calculated components. - Draw the resultant from the origin (start) to the tip of the last vector. --- **[If you need a visual illustration, please ask for a diagram; otherwise, use these values to plot on your graph paper.]**