Math / Algebra

QuestionUse the figure to evaluate $\mathbf{a}+\mathbf{b}, \mathbf{a}-\mathbf{b}$ , and $-\mathbf{a}$ . $a+b=\langle[$ $\square$ $\square$ $7\rangle$

Studdy Solution

STEP 1

What is this asking? We're given two vectors, $\mathbf{a}$ and $\mathbf{b}$ , visually on a graph, and we need to find $\mathbf{a} + \mathbf{b}$ , $\mathbf{a} - \mathbf{b}$ , and $-\mathbf{a}$ . Watch out! Don't mix up adding and subtracting vectors!
Remember, adding vectors is like following a path, while subtracting is like finding the difference in their positions.
Also, be careful with the signs when working with the components of the vectors.

STEP 2

1. Find the components of vector a
2. Find the components of vector b
3. Calculate a + b
4. Calculate a - b
5. Calculate -a

STEP 3

Vector $\mathbf{a}$ starts at the origin $(0, 0)$ and goes to the point $(-3, 4)$ .
So, the components of $\mathbf{a}$ are simply $\langle -3, 4 \rangle$ .
It's that easy!

STEP 4

Vector $\mathbf{b}$ starts at the tip of $\mathbf{a}$ , which is at $(-3, 4)$ , and goes to the point $(-2, 2)$ .
To find the components of $\mathbf{b}$ , we subtract the starting point from the ending point: $(-2 - (-3), 2 - 4) = (1, -2)$ .
So, $\mathbf{b} = \langle 1, -2 \rangle$ .

STEP 5

To add vectors, we simply add their corresponding components.
So, $\mathbf{a} + \mathbf{b} = \langle -3 + 1, 4 + (-2) \rangle = \langle -2, 2 \rangle$ .
Boom!

STEP 6

Subtracting vectors is just like adding, but we subtract the components instead.
So, $\mathbf{a} - \mathbf{b} = \langle -3 - 1, 4 - (-2) \rangle = \langle -4, 6 \rangle$ .
Nailed it!

STEP 7

To find the negative of a vector, we simply multiply each component by $-1$ .
So, $-\mathbf{a} = \langle -1 \cdot (-3), -1 \cdot 4 \rangle = \langle 3, -4 \rangle$ .
Awesome!

STEP 8

$\mathbf{a} + \mathbf{b} = \langle -2, 2 \rangle$ $\mathbf{a} - \mathbf{b} = \langle -4, 6 \rangle$ $-\mathbf{a} = \langle 3, -4 \rangle$

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QuestionUse the figure to evaluate $\mathbf{a}+\mathbf{b}, \mathbf{a}-\mathbf{b}$ , and $-\mathbf{a}$ . $a+b=\langle[$ $\square$ $\square$ $7\rangle$

Studdy Solution

STEP 1

STEP 2

1. Find the components of vector a
2. Find the components of vector b
3. Calculate a + b
4. Calculate a - b
5. Calculate -a

STEP 3

Vector $\mathbf{a}$ starts at the origin $(0, 0)$ and goes to the point $(-3, 4)$ .
So, the components of $\mathbf{a}$ are simply $\langle -3, 4 \rangle$ .
It's that easy!

STEP 4

Vector $\mathbf{b}$ starts at the tip of $\mathbf{a}$ , which is at $(-3, 4)$ , and goes to the point $(-2, 2)$ .
To find the components of $\mathbf{b}$ , we subtract the starting point from the ending point: $(-2 - (-3), 2 - 4) = (1, -2)$ .
So, $\mathbf{b} = \langle 1, -2 \rangle$ .

STEP 5

To add vectors, we simply add their corresponding components.
So, $\mathbf{a} + \mathbf{b} = \langle -3 + 1, 4 + (-2) \rangle = \langle -2, 2 \rangle$ .
Boom!

STEP 6

Subtracting vectors is just like adding, but we subtract the components instead.
So, $\mathbf{a} - \mathbf{b} = \langle -3 - 1, 4 - (-2) \rangle = \langle -4, 6 \rangle$ .
Nailed it!

STEP 7

To find the negative of a vector, we simply multiply each component by $-1$ .
So, $-\mathbf{a} = \langle -1 \cdot (-3), -1 \cdot 4 \rangle = \langle 3, -4 \rangle$ .
Awesome!

STEP 8

$\mathbf{a} + \mathbf{b} = \langle -2, 2 \rangle$ $\mathbf{a} - \mathbf{b} = \langle -4, 6 \rangle$ $-\mathbf{a} = \langle 3, -4 \rangle$

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QuestionUse the figure to evaluate a+b,a−b\mathbf{a}+\mathbf{b}, \mathbf{a}-\mathbf{b}a+b,a−b, and −a-\mathbf{a}−a. a+b=⟨[a+b=\langle[a+b=⟨[ □\square□ □\square□ 7⟩7\rangle7⟩

Studdy Solution

STEP 1

STEP 2

1. Find the components of vector a2. Find the components of vector b3. Calculate a + b4. Calculate a - b5. Calculate -a

STEP 3

Vector a\mathbf{a}a starts at the origin (0,0)(0, 0)(0,0) and goes to the point (−3,4)(-3, 4)(−3,4).So, the components of a\mathbf{a}a are simply ⟨−3,4⟩\langle -3, 4 \rangle⟨−3,4⟩.It's that easy!

STEP 4

STEP 5

To add vectors, we simply add their corresponding components.So, a+b=⟨−3+1,4+(−2)⟩=⟨−2,2⟩\mathbf{a} + \mathbf{b} = \langle -3 + 1, 4 + (-2) \rangle = \langle -2, 2 \ranglea+b=⟨−3+1,4+(−2)⟩=⟨−2,2⟩.Boom!

STEP 6

Subtracting vectors is just like adding, but we subtract the components instead.So, a−b=⟨−3−1,4−(−2)⟩=⟨−4,6⟩\mathbf{a} - \mathbf{b} = \langle -3 - 1, 4 - (-2) \rangle = \langle -4, 6 \ranglea−b=⟨−3−1,4−(−2)⟩=⟨−4,6⟩.Nailed it!

STEP 7

To find the negative of a vector, we simply multiply each component by −1-1−1.So, −a=⟨−1⋅(−3),−1⋅4⟩=⟨3,−4⟩-\mathbf{a} = \langle -1 \cdot (-3), -1 \cdot 4 \rangle = \langle 3, -4 \rangle−a=⟨−1⋅(−3),−1⋅4⟩=⟨3,−4⟩.Awesome!

STEP 8

a+b=⟨−2,2⟩\mathbf{a} + \mathbf{b} = \langle -2, 2 \ranglea+b=⟨−2,2⟩ a−b=⟨−4,6⟩\mathbf{a} - \mathbf{b} = \langle -4, 6 \ranglea−b=⟨−4,6⟩ −a=⟨3,−4⟩-\mathbf{a} = \langle 3, -4 \rangle−a=⟨3,−4⟩

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QuestionUse the figure to evaluate $\mathbf{a}+\mathbf{b}, \mathbf{a}-\mathbf{b}$ , and $-\mathbf{a}$ . $a+b=\langle[$ $\square$ $\square$ $7\rangle$

1. Find the components of vector a
2. Find the components of vector b
3. Calculate a + b
4. Calculate a - b
5. Calculate -a

Vector $\mathbf{a}$ starts at the origin $(0, 0)$ and goes to the point $(-3, 4)$ .
So, the components of $\mathbf{a}$ are simply $\langle -3, 4 \rangle$ .
It's that easy!

To add vectors, we simply add their corresponding components.
So, $\mathbf{a} + \mathbf{b} = \langle -3 + 1, 4 + (-2) \rangle = \langle -2, 2 \rangle$ .
Boom!

Subtracting vectors is just like adding, but we subtract the components instead.
So, $\mathbf{a} - \mathbf{b} = \langle -3 - 1, 4 - (-2) \rangle = \langle -4, 6 \rangle$ .
Nailed it!

To find the negative of a vector, we simply multiply each component by $-1$ .
So, $-\mathbf{a} = \langle -1 \cdot (-3), -1 \cdot 4 \rangle = \langle 3, -4 \rangle$ .
Awesome!

$\mathbf{a} + \mathbf{b} = \langle -2, 2 \rangle$ $\mathbf{a} - \mathbf{b} = \langle -4, 6 \rangle$ $-\mathbf{a} = \langle 3, -4 \rangle$