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.

Verified step by step guidance

1. Step 1: Break down each segment of the professor's journey into vector components. The first segment is 3.25 km north, which can be represented as a vector with components (0, 3.25) in the Cartesian coordinate system, where the positive y-axis points north.
2. Step 2: Represent the second segment, 2.20 km west, as a vector with components (-2.20, 0), where the negative x-axis points west.
3. Step 3: Represent the third segment, 1.50 km south, as a vector with components (0, -1.50), where the negative y-axis points south.
4. Step 4: Add the vector components from each segment to find the resultant vector. Sum the x-components: 0 + (-2.20) + 0 = -2.20 km. Sum the y-components: 3.25 + 0 + (-1.50) = 1.75 km.
5. Step 5: Calculate the magnitude of the resultant displacement vector using the Pythagorean theorem: \( \sqrt{(-2.20)^2 + (1.75)^2} \). Determine the direction by calculating the angle \( \theta \) with respect to the west direction using \( \tan^{-1}(\frac{1.75}{2.20}) \).

Key Concepts

Vector Addition

Vector addition involves combining vectors to determine a resultant vector. In this context, the professor's movements north, west, and south are vectors that need to be added. The vectors are added by placing them head to tail, and the resultant vector is drawn from the tail of the first vector to the head of the last vector.

Components of a Vector

The method of components involves breaking down vectors into their horizontal and vertical components. For this problem, the north and south movements affect the vertical component, while the west movement affects the horizontal component. By calculating these components, we can determine the overall displacement in each direction and find the resultant vector.

Pythagorean Theorem and Trigonometry

The Pythagorean theorem is used to calculate the magnitude of the resultant vector from its components. Once the horizontal and vertical components are known, the magnitude is found using the formula: magnitude = √(horizontal² + vertical²). Trigonometry, specifically the tangent function, helps find the direction of the resultant vector by calculating the angle relative to a reference direction.