Question

use the figure to evaluate a + b, a - b, and - a.
a + b = (\square, \square)
a - b = (\square, \square)

• a = (\square, \square)

Explanation:

Step1: Determine vectors \(\mathbf{a}\) and \(\mathbf{b}\)

From the grid, vector \(\mathbf{a}\) has components: starting at the origin, moving \(0\) in \(x\)-direction and \(5\) in \(y\)-direction? Wait, no, looking at the graph: vector \(\mathbf{a}\) is from origin \((0,0)\) to \((-1, 5)\)? Wait, no, let's check the grid. Each square is 1 unit. Vector \(\mathbf{a}\): from (0,0) to (-1, 5)? Wait, no, the blue arrow for \(\mathbf{a}\): x-component: from 0 to -1? Wait, no, maybe I misread. Wait, vector \(\mathbf{a}\): let's see, the tail at origin, head at \((-1, 5)\)? Wait, no, maybe \(\mathbf{a}\) is \(\langle -1, 5
angle\) and \(\mathbf{b}\) is \(\langle 3, 5
angle\)? Wait, let's check:

Wait, vector \(\mathbf{a}\): from (0,0) to (-1, 5)? Wait, no, the x-axis: left is negative, right is positive. So vector \(\mathbf{a}\): x-component: -1 (since it moves 1 left), y-component: 5 (moves 5 up). Vector \(\mathbf{b}\): x-component: 3 (moves 3 right), y-component: 5 (moves 5 up). So \(\mathbf{a} = \langle -1, 5
angle\), \(\mathbf{b} = \langle 3, 5
angle\).

Step2: Calculate \(\mathbf{a} + \mathbf{b}\)

To add vectors, add their corresponding components:
\(x\)-component: \(-1 + 3 = 2\)
\(y\)-component: \(5 + 5 = 10\)
So \(\mathbf{a} + \mathbf{b} = \langle 2, 10
angle\).

Step3: Calculate \(\mathbf{a} - \mathbf{b}\)

Subtract corresponding components:
\(x\)-component: \(-1 - 3 = -4\)
\(y\)-component: \(5 - 5 = 0\)
So \(\mathbf{a} - \mathbf{b} = \langle -4, 0
angle\).

Step4: Calculate \(-\mathbf{a}\)

Negate each component of \(\mathbf{a}\):
\(x\)-component: \(-(-1) = 1\)
\(y\)-component: \(-5\)
So \(-\mathbf{a} = \langle 1, -5
angle\).

Answer:

\(a + b = \langle 2, 10
angle\)
\(a - b = \langle -4, 0
angle\)
\(-a = \langle 1, -5
angle\)