Question

A disoriented physics professor drives 3.25 km north, then 2.20 km west, and then 1.50 km south. Find the magnitude and direction of the resultant displacement, using the method of components. In a vector-addition diagram (roughly to scale), show that the resultant displacement found from your diagram is in qualitative agreement with the result you obtained by using the method of components.

Accepted Answer

Begin by identifying the vectors involved in the problem. The professor drives 3.25 km north, 2.20 km west, and 1.50 km south. These can be represented as vectors: $$ \mathbf{A} = 3.25 	ext{ km north} $$, $$ \mathbf{B} = 2.20 	ext{ km west} $$, and $$ \mathbf{C} = 1.50 	ext{ km south} . $$Convert each vector into its components. For vector $$ \mathbf{A} $$, the north direction corresponds to the positive y-axis, so $$ \mathbf{A} = (0, 3.25) . $$For vector $$ \mathbf{B} $$, the west direction corresponds to the negative x-axis, so $$ \mathbf{B} = (-2.20, 0) . $$For vector $$ \mathbf{C} $$, the south direction corresponds to the negative y-axis, so $$ \mathbf{C} = (0, -1.50) . $$Add the components of the vectors to find the resultant vector $$ \mathbf{R} . $$The x-component of $$ \mathbf{R}  is $$the sum of the x-components of $$ \mathbf{A} $$, $$ \mathbf{B} $$, and $$ \mathbf{C} $$: $$ R_x = 0 + (-2.20) + 0 = -2.20 . $$The y-component of $$ \mathbf{R}  is $$the sum of the y-components: $$ R_y = 3.25 + 0 + (-1.50) = 1.75 . $$Thus, $$ \mathbf{R} = (-2.20, 1.75) . $$Calculate the magnitude of the resultant displacement vector $$ \mathbf{R} $$ using the Pythagorean theorem: $$ |\mathbf{R}| = \sqrt{R_x^2 + R_y^2} = \sqrt{(-2.20)^2 + (1.75)^2} . $$Determine the direction of the resultant displacement vector $$ \mathbf{R}  by $$calculating the angle $$ 	heta $$ with respect to the negative x-axis using the tangent function: $$ 	heta = 	an^{-1}\left(\frac{R_y}{R_x}ight) = 	an^{-1}\left(\frac{1.75}{-2.20}ight) . $$This angle will give the direction of the displacement vector relative to the west direction.

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Concetti chiave

Vector Addition

Components of Vectors

Resultant Displacement