Suppose you first walk 14m
in a direction 20∘
west of north, as represented by vector A⃗
in the drawing. Then you walk 23m
in a direction 40∘
south of west as represented by vector B⃗
. Your final position is represented by vector R⃗
with the direction θ
south of west. Associate positive x
with east and positive y
with north.

Question

Suppose you first walk 14m
 in a direction 20∘
 west of north, as represented by vector A⃗ 
 in the drawing. Then you walk 23m
 in a direction 40∘
 south of west as represented by vector B⃗ 
. Your final position is represented by vector R⃗ 
 with the direction θ
 south of west. Associate positive x
 with east and positive y
 with north.

Accepted Answer

∻Solution∻ ‖ a ⋮ First, we need to resolve vector A into its components. The angle is 20° west of north, which means we can use trigonometry to find the x and y components: $$A_x = 14 \sin(20°)$$ $$A_y = 14 \cos(20°)$$ ‖ ‖ b ⋮ Next, we resolve vector B into its components. The angle is 40° south of west, so we have: $$B_x = -23 \cos(40°)$$ $$B_y = -23 \sin(40°)$$ ‖ ‖ c ⋮ Now, we can find the resultant vector R by adding the components of A and B: $$R_x = A_x + B_x$$ $$R_y = A_y + B_y$$ ‖ ‖ d ⋮ Finally, we can find the magnitude and direction of vector R. The magnitude is given by: $$|R| = \sqrt{R_x^2 + R_y^2}$$ The direction θ south of west can be found using: $$ heta = an^{-1}\left(\frac{|R_y|}{|R_x|} ight)$$ ‖ ∻Answer∻ ⚹ [Insert final answer here] ⚹ ∻Key Concept∻ ⚹ Vector Addition: The resultant vector is found by adding the components of the individual vectors. The direction and magnitude can be calculated using trigonometric functions. ⚹ ∻Explanation∻ ⚹ By resolving the vectors into their components and then summing them, we can determine the final position and direction of the resultant vector. This method is essential in kinematics for analyzing motion in two dimensions. ⚹

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