Question

use the figure to evaluate $a + b$, $a - b$, and $-a$.
$a + b=langle 2,9
angle$
$a - b=langle -4,-1
angle$
$-a=langlesquare,square
angle$

Explanation:

Step1: Identify vector a

From the figure, vector \( \mathbf{a} \) has components. Let's assume vector \( \mathbf{a} = \langle -1, 4 angle \) (by looking at the grid: from origin, moving left 1 unit (x-component -1) and up 4 units (y-component 4)). Wait, but let's check the given \( \mathbf{a + b} = \langle 2, 9 angle \) and \( \mathbf{a - b} = \langle -4, -1 angle \). Alternatively, to find \( -\mathbf{a} \), we know that for a vector \( \mathbf{a} = \langle x, y angle \), \( -\mathbf{a} = \langle -x, -y angle \). Let's find \( \mathbf{a} \) first. Let \( \mathbf{a} = \langle x_1, y_1 angle \), \( \mathbf{b} = \langle x_2, y_2 angle \). Then \( \mathbf{a + b} = \langle x_1 + x_2, y_1 + y_2 angle = \langle 2, 9 angle \), \( \mathbf{a - b} = \langle x_1 - x_2, y_1 - y_2 angle = \langle -4, -1 angle \). Solving these two equations: adding them, \( 2\mathbf{a} = \langle 2 - 4, 9 - 1 angle = \langle -2, 8 angle \), so \( \mathbf{a} = \langle -1, 4 angle \). Therefore, \( -\mathbf{a} = \langle 1, -4 angle \)? Wait, no, wait. Wait, maybe from the figure, vector \( \mathbf{a} \) is from the origin to (-1, 4)? Wait, no, looking at the figure: the blue arrow for \( \mathbf{a} \) is at x=-1 (since it's left of y-axis) and y=4 (from origin up 4). Wait, but let's check the grid. The y-axis is at the center. The vector \( \mathbf{a} \): the tail is at origin, head at (-1, 4)? Wait, no, the figure shows vector \( \mathbf{a} \) with head at (-1, 4) (since x=-1, y=4) and vector \( \mathbf{b} \) with head at (3, 5)? Wait, no, given \( \mathbf{a + b} = (2,9) \) and \( \mathbf{a - b} = (-4, -1) \). Let's solve for \( \mathbf{a} \):

Let \( \mathbf{a} = (x, y) \), \( \mathbf{b} = (u, v) \)

\( x + u = 2 \)

\( y + v = 9 \)

\( x - u = -4 \)

\( y - v = -1 \)

Adding the first and third equations: \( 2x = 2 - 4 = -2 \implies x = -1 \)

Adding the second and fourth equations: \( 2y = 9 - 1 = 8 \implies y = 4 \)

So \( \mathbf{a} = (-1, 4) \), therefore \( -\mathbf{a} = (1, -4) \)? Wait, no, wait, maybe I made a mistake. Wait, the vector \( \mathbf{a} \) in the figure: looking at the grid, the x-coordinate of \( \mathbf{a} \)'s head is -1 (since it's one unit left of y-axis) and y-coordinate is 4 (four units up). So \( \mathbf{a} = \langle -1, 4 angle \), so \( -\mathbf{a} = \langle 1, -4 angle \)? But wait, let's check with the given \( \mathbf{a + b} = (2,9) \). If \( \mathbf{a} = (-1, 4) \), then \( \mathbf{b} = (2 - (-1), 9 - 4) = (3, 5) \). Then \( \mathbf{a - b} = (-1 - 3, 4 - 5) = (-4, -1) \), which matches. So \( \mathbf{a} = (-1, 4) \), so \( -\mathbf{a} = (1, -4) \)? Wait, no, wait, the problem's \( \mathbf{a + b} \) is given as (2,9), \( \mathbf{a - b} \) as (-4, -1). So \( \mathbf{a} = (-1, 4) \), so \( -\mathbf{a} = (1, -4) \)? Wait, but maybe the vector \( \mathbf{a} \) is ( -1, 4 )? Wait, no, maybe I messed up. Wait, let's look at the figure again. The vector \( \mathbf{a} \): the tail is at the origin, head at ( -1, 4 )? Wait, the x-axis: from -5 to 5, y-axis from -5 to 5. The vector \( \mathbf{a} \) is blue, with head at x=-1, y=4 (since it's one grid left of y-axis and four grids up). So \( \mathbf{a} = \langle -1, 4 angle \). Therefore, \( -\mathbf{a} = \langle 1, -4 angle \)? Wait, but let's check with the given. Wait, maybe the vector \( \mathbf{a} \) is ( -1, 4 ), so \( -\mathbf{a} = (1, -4) \). Wait, but let's confirm. Alternatively, maybe the vector \( \mathbf{a} \) is ( -1, 4 ), so \( -\mathbf{a} = (1, -4) \). Wait, but let's check the grid. Wait, the vector \( \mathbf{a} \): from the origin (the vertex of the two vectors), moving to (-1, 4) (x=-1, y=4). So \( \mathbf{a} = \langle -1, 4 angle \), so \( -\mathbf{a} = \langle 1, -4 angle \). Wait, but let's check with the given \( \mathbf{a + b} = (2,9) \). If \( \mathbf{a} = (-1, 4) \), then \( \mathbf{b} = (2 - (-1), 9 - 4) = (3, 5) \). Then \( \mathbf{a - b} = (-1 - 3, 4 - 5) = (-4, -1) \), which matches. So \( -\mathbf{a} = (1, -4) \). Wait, but maybe I made a mistake. Wait, the problem's \( -\mathbf{a} \): let's think again. For a vector \( \mathbf{a} = \langle x, y angle \), \( -\mathbf{a} = \langle -x, -y angle \). So if \( \mathbf{a} = \langle -1, 4 angle \), then \( -\mathbf{a} = \langle 1, -4 angle \). Yes, that's correct.

Step2: Calculate -a

Given \( \mathbf{a} = \langle -1, 4 angle \) (from solving the system or from the figure), then \( -\mathbf{a} = \langle -(-1), -4 angle = \langle 1, -4 angle \). Wait, but let's check the grid again. Wait, maybe the vector \( \mathbf{a} \) is ( -1, 4 ), so \( -\mathbf{a} = (1, -4) \).

Answer:

\( -\mathbf{a} = \langle 1, -4
angle \)