Find the resultant vector (magnitude and direction) by adding the following set of vectors: A – 7.7m due South B = 4.5m at 60° South of East C = 7.4m due East State the direction as an angle between 0 and 90 degrees north of east, north of west, south of east, or south of west.
The Correct Answer and Explanation is:
To find the resultant vector when adding vectors A, B, and C, we need to break each vector into its components along the x-axis (East-West) and y-axis (North-South). Then, we can sum up the components in each direction to get the resultant vector. Finally, we’ll find its magnitude and direction.
Step 1: Resolve each vector into components
Vector A (7.7 m due South):
- x-component: 0 (since it only points south, there is no East-West component)
- y-component: -7.7 m (negative because it points south)
Vector B (4.5 m at 60° South of East):
- x-component: Bx=4.5×cos(60∘)=4.5×0.5=2.25B_x = 4.5 \times \cos(60^\circ) = 4.5 \times 0.5 = 2.25Bx=4.5×cos(60∘)=4.5×0.5=2.25 m (eastward)
- y-component: By=4.5×sin(60∘)=4.5×0.866=3.897B_y = 4.5 \times \sin(60^\circ) = 4.5 \times 0.866 = 3.897By=4.5×sin(60∘)=4.5×0.866=3.897 m (southward)
Vector C (7.4 m due East):
- x-component: 7.4 m (eastward)
- y-component: 0 (since it only points east, there is no North-South component)
Step 2: Sum up the components
x-components (East-West):
Rx=Ax+Bx+Cx=0+2.25+7.4=9.65 mR_x = A_x + B_x + C_x = 0 + 2.25 + 7.4 = 9.65 \, \text{m}Rx=Ax+Bx+Cx=0+2.25+7.4=9.65m
y-components (North-South):
Ry=Ay+By+Cy=−7.7+(−3.897)+0=−11.597 mR_y = A_y + B_y + C_y = -7.7 + (-3.897) + 0 = -11.597 \, \text{m}Ry=Ay+By+Cy=−7.7+(−3.897)+0=−11.597m
Step 3: Find the magnitude of the resultant vector
The magnitude of the resultant vector RRR is given by the Pythagorean theorem: R=Rx2+Ry2=(9.65)2+(−11.597)2R = \sqrt{R_x^2 + R_y^2} = \sqrt{(9.65)^2 + (-11.597)^2}R=Rx2+Ry2=(9.65)2+(−11.597)2 R=93.1225+134.4328=227.5553≈15.08 mR = \sqrt{93.1225 + 134.4328} = \sqrt{227.5553} \approx 15.08 \, \text{m}R=93.1225+134.4328=227.5553≈15.08m
Step 4: Find the direction of the resultant vector
The direction is the angle θ\thetaθ between the resultant vector and the positive x-axis (East), measured counterclockwise. We can use the inverse tangent function: θ=tan−1(RyRx)=tan−1(−11.5979.65)\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{-11.597}{9.65}\right)θ=tan−1(RxRy)=tan−1(9.65−11.597) θ=tan−1(−1.20)≈−50.19∘\theta = \tan^{-1}(-1.20) \approx -50.19^\circθ=tan−1(−1.20)≈−50.19∘
Since the angle is negative, it means the vector is pointing south of east. To express the angle as a positive value: θ=50.19∘ south of east\theta = 50.19^\circ \, \text{south of east}θ=50.19∘south of east
Final Answer:
- Magnitude: 15.08 m
- Direction: 50.19° south of east
