Problem

Three vectors labeled $\mathbf{A}, \mathbf{B}$ and $\mathbf{R}$ are on an $x y$, coordinate plane. Vector $\mathbf{A}$ begins from the origin and extends to the left along the negative $x$-axis. The tail of vector $\mathbf{B}$ begins at the head of vector $\mathbf{A}$ and extends straight up in the positive $y$-direction. The tail of vector $\mathbf{R}$ is located at the origin and the head of vector $\mathbf{R}$ is located at the same position as the head of vector B. Next to the coordinate system, the equation $\mathbf{A}+\mathbf{B}=\mathbf{R}$ and a coordinate system depicting North, East, South, and West with an arrow facing upward toward the north are also shown. What is the compass direction of a line connecting your starting point to your final position measured in degrees west of north? $+$

Solution

Step 1 :The problem describes a situation where we start at the origin of a coordinate system, move along vector A (which is along the negative x-axis), and then move along vector B (which is along the positive y-axis). The final position is the head of vector R, which is the result of the vector addition of A and B.

Step 2 :The compass direction of a line connecting the starting point (origin) to the final position (head of R) is the direction of vector R. Since vector R is the result of moving left along the x-axis (west) and then up along the y-axis (north), the direction of vector R is northwest.

Step 3 :However, the question asks for the direction measured in degrees west of north. This means we need to find the angle between the positive y-axis (north) and vector R, measured in the clockwise direction.

Step 4 :Since vector A is along the negative x-axis and vector B is along the positive y-axis, the angle between vector A and vector B is 90 degrees. Therefore, the angle between the positive y-axis (north) and vector R is also 90 degrees.

Step 5 :Final Answer: The compass direction of a line connecting the starting point to the final position, measured in degrees west of north, is \(\boxed{90}\).

From Solvely APP
Source: https://solvelyapp.com/problems/16900/

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