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Math  /  Algebra

QuestionQuestion 5 of 18 This test: 18 point(s) possible This question: 1 point(s) possible
Use the figure to evaluate a+b,a−b\mathbf{a}+\mathbf{b}, \mathbf{a}-\mathbf{b}, and −a-\mathbf{a} a+b=⟨□,□⟩a−b=⟨□,□⟩−a=⟨□,□⟩\begin{array}{c} \mathrm{a}+\mathrm{b}=\langle\square, \square\rangle \\ \mathrm{a}-\mathrm{b}=\langle\square, \square\rangle \\ -\mathbf{a}=\langle\square, \square\rangle \end{array}

Studdy Solution

STEP 1

1. Vectors a\mathbf{a} and b\mathbf{b} are given in component form based on their terminal points.
2. Vector addition and subtraction follow the rules of adding and subtracting corresponding components.
3. Negating a vector involves negating each of its components.

STEP 2

1. Determine the components of vectors a\mathbf{a} and b\mathbf{b}.
2. Calculate a+b\mathbf{a} + \mathbf{b}.
3. Calculate a−b\mathbf{a} - \mathbf{b}.
4. Calculate −a-\mathbf{a}.

STEP 3

Identify the components of vectors a\mathbf{a} and b\mathbf{b} from the given points: - a=⟨3,2⟩\mathbf{a} = \langle 3, 2 \rangle - b=⟨1,4⟩\mathbf{b} = \langle 1, 4 \rangle

STEP 4

Calculate a+b\mathbf{a} + \mathbf{b} by adding the corresponding components of a\mathbf{a} and b\mathbf{b}: a+b=⟨3+1,2+4⟩=⟨4,6⟩ \mathbf{a} + \mathbf{b} = \langle 3 + 1, 2 + 4 \rangle = \langle 4, 6 \rangle

STEP 5

Calculate a−b\mathbf{a} - \mathbf{b} by subtracting the corresponding components of b\mathbf{b} from a\mathbf{a}: a−b=⟨3−1,2−4⟩=⟨2,−2⟩ \mathbf{a} - \mathbf{b} = \langle 3 - 1, 2 - 4 \rangle = \langle 2, -2 \rangle

STEP 6

Calculate −a-\mathbf{a} by negating each component of a\mathbf{a}: −a=⟨−3,−2⟩ -\mathbf{a} = \langle -3, -2 \rangle
The evaluated vectors are: a+b=⟨4,6⟩\mathbf{a} + \mathbf{b} = \langle 4, 6 \rangle a−b=⟨2,−2⟩\mathbf{a} - \mathbf{b} = \langle 2, -2 \rangle −a=⟨−3,−2⟩-\mathbf{a} = \langle -3, -2 \rangle

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