Question

type the correct answers in each box.
the graph shows vectors w and z.
graph of vectors w (purple, from origin to (-4,1)) and z (red, from origin to (3,4))
the resulting vector for ( mathbf{w} - mathbf{z} ) is ( langle square, square
angle ), and ( mathbf{z} - mathbf{w} ) is ( langle square, square
angle ).

Explanation:

Step1: Determine coordinates of vectors

First, find the coordinates of vector \( \mathbf{W} \) and \( \mathbf{Z} \). From the graph, vector \( \mathbf{W} \) goes from the origin \((0,0)\) to \((-4, 1)\)? Wait, no, looking at the graph: Vector \( \mathbf{W} \) has its tip at \((-4, 1)\)? Wait, no, the purple vector \( \mathbf{W} \) starts at the origin? Wait, no, the graph shows vector \( \mathbf{W} \) (purple) with initial point at the origin? Wait, no, the x-axis: the left end of \( \mathbf{W} \) is at \((-4, 1)\) and it goes to the origin? Wait, no, the vector \( \mathbf{W} \): let's check the coordinates. The vector \( \mathbf{W} \) has components: from the origin (0,0) to (-4, 1)? Wait, no, the purple vector \( \mathbf{W} \) has its terminal point at (-4, 1)? Wait, no, the x-axis is from -4 to 3, y-axis from -5 to 5. Wait, vector \( \mathbf{W} \): the initial point is the origin (0,0), and the terminal point is (-4, 1)? Wait, no, the purple vector \( \mathbf{W} \) is going from (0,0) to (-4, 1)? Wait, no, looking at the graph: the purple vector \( \mathbf{W} \) has a terminal point at (-4, 1)? Wait, no, the x-coordinate: the leftmost point of \( \mathbf{W} \) is at x=-4, y=1, and it goes to the origin (0,0). So vector \( \mathbf{W} \) is \( \langle -4, 1 angle \)? Wait, no, the vector from origin to (-4,1) is \( \langle -4, 1 angle \). Then vector \( \mathbf{Z} \) (red) goes from origin (0,0) to (3, 4)? Wait, the red vector \( \mathbf{Z} \) has terminal point at (3, 4). So \( \mathbf{Z} = \langle 3, 4 angle \), \( \mathbf{W} = \langle -4, 1 angle \)? Wait, no, wait: the purple vector \( \mathbf{W} \): let's check the direction. Wait, maybe I got it wrong. Wait, the vector \( \mathbf{W} \) is from the origin to (-4, 1)? No, the purple vector \( \mathbf{W} \) has a terminal point at (-4, 1)? Wait, no, the x-axis: the left end of \( \mathbf{W} \) is at (-4, 1), and it goes to the origin (0,0). So vector \( \mathbf{W} \) is \( \langle -4, 1 angle \) (from origin to (-4,1)). Vector \( \mathbf{Z} \) is from origin to (3, 4), so \( \mathbf{Z} = \langle 3, 4 angle \).

Step2: Calculate \( \mathbf{W} - \mathbf{Z} \)

To find \( \mathbf{W} - \mathbf{Z} \), we subtract the components of \( \mathbf{Z} \) from \( \mathbf{W} \). So if \( \mathbf{W} = \langle w_x, w_y angle \) and \( \mathbf{Z} = \langle z_x, z_y angle \), then \( \mathbf{W} - \mathbf{Z} = \langle w_x - z_x, w_y - z_y angle \). Wait, no: vector subtraction \( \mathbf{W} - \mathbf{Z} = \mathbf{W} + (-\mathbf{Z}) \), so components are \( (w_x - z_x, w_y - z_y) \). Wait, but maybe I mixed up the vectors. Wait, maybe the vector \( \mathbf{W} \) is from the origin to (-4, 0.5)? Wait, looking at the graph again: the purple vector \( \mathbf{W} \) has a terminal point at (-4, 1)? Wait, the y-coordinate at x=-4 is 1, and it goes to the origin. So \( \mathbf{W} = \langle -4, 1 angle \), \( \mathbf{Z} = \langle 3, 4 angle \) (since the red vector goes to (3,4)). Then \( \mathbf{W} - \mathbf{Z} = \langle -4 - 3, 1 - 4 angle = \langle -7, -3 angle \). Wait, but that seems off. Wait, maybe the vector \( \mathbf{W} \) is from (0,0) to (-4, 0.5)? Wait, the y-axis: the purple vector is at y=1? Wait, the grid lines: each grid is 1 unit. So the purple vector \( \mathbf{W} \) has terminal point at (-4, 1), and the red vector \( \mathbf{Z} \) has terminal point at (3, 4). Then \( \mathbf{W} = \langle -4, 1 angle \), \( \mathbf{Z} = \langle 3, 4 angle \). Then \( \mathbf{W} - \mathbf{Z} = (-4 - 3, 1 - 4) = (-7, -3) \). And \( \mathbf{Z} - \mathbf{W} = (3 - (-4), 4 - 1) = (7, 3) \). Wait, but let's check again. Wait, maybe the vector \( \mathbf{W} \) is from (0,0) to (-4, 0.5)? No, the graph shows the purple vector at y=1. Alternatively, maybe the vector \( \mathbf{W} \) is \( \langle -4, 0.5 angle \), but the problem is to type the correct answers. Wait, maybe I made a mistake. Wait, the vector \( \mathbf{W} \): from the origin (0,0) to (-4, 1) – so components (-4, 1). Vector \( \mathbf{Z} \): from origin (0,0) to (3, 4) – components (3, 4). Then \( \mathbf{W} - \mathbf{Z} = (-4 - 3, 1 - 4) = (-7, -3) \). \( \mathbf{Z} - \mathbf{W} = (3 - (-4), 4 - 1) = (7, 3) \). Wait, but let's confirm with vector subtraction. The formula for vector subtraction: if \( \mathbf{a} = (a_1, a_2) \), \( \mathbf{b} = (b_1, b_2) \), then \( \mathbf{a} - \mathbf{b} = (a_1 - b_1, a_2 - b_2) \). So yes, that's correct.

Step3: Verify the calculations

Check the components again. For \( \mathbf{W} \): starting at (0,0), ending at (-4, 1) – so change in x is -4, change in y is 1. For \( \mathbf{Z} \): starting at (0,0), ending at (3, 4) – change in x is 3, change in y is 4. So \( \mathbf{W} = \langle -4, 1 angle \), \( \mathbf{Z} = \langle 3, 4 angle \). Then \( \mathbf{W} - \mathbf{Z} = (-4 - 3, 1 - 4) = (-7, -3) \). \( \mathbf{Z} - \mathbf{W} = (3 - (-4), 4 - 1) = (7, 3) \).

Answer:

For \( \mathbf{W} - \mathbf{Z} \): \(-7\), \(-3\); for \( \mathbf{Z} - \mathbf{W} \): \(7\), \(3\)