Question

1. sketch \\(vec{v}\\), \\(3vec{v}\\), \\(vec{w}\\), \\(2vec{w}\\), \\(vec{v} + vec{w}\\), and \\(vec{v} - vec{w}\\) for the following:
2. \\(vec{v} = langle 1, 0

angle\\), \\(vec{w} = langle 0, 1
angle\\).

1. \\(vec{v} = langle -1, 1

angle\\), \\(vec{w} = langle 1, 2
angle\\).

1. \\(vec{v} = langle 1, -2

angle\\), \\(vec{w} = langle 3, 1
angle\\).

Explanation:

For Part 1: $\vec{v} = \langle 1,0

angle$, $\vec{w} = \langle 0,1
angle$

Step1: Identify base vectors

$\vec{v} = \langle 1,0
angle$, $\vec{w} = \langle 0,1
angle$

Step2: Compute scalar multiples

$3\vec{v} = 3\langle 1,0
angle = \langle 3,0
angle$
$2\vec{w} = 2\langle 0,1
angle = \langle 0,2
angle$

Step3: Compute vector sum

$\vec{v} + \vec{w} = \langle 1+0, 0+1
angle = \langle 1,1
angle$

Step4: Compute vector difference

$\vec{v} - \vec{w} = \langle 1-0, 0-1
angle = \langle 1,-1
angle$

---

For Part 2: $\vec{v} = \langle -1,1

angle$, $\vec{w} = \langle 1,2
angle$

Step1: Identify base vectors

$\vec{v} = \langle -1,1
angle$, $\vec{w} = \langle 1,2
angle$

Step2: Compute scalar multiples

$3\vec{v} = 3\langle -1,1
angle = \langle -3,3
angle$
$2\vec{w} = 2\langle 1,2
angle = \langle 2,4
angle$

Step3: Compute vector sum

$\vec{v} + \vec{w} = \langle -1+1, 1+2
angle = \langle 0,3
angle$

Step4: Compute vector difference

$\vec{v} - \vec{w} = \langle -1-1, 1-2
angle = \langle -2,-1
angle$

---

For Part 3: $\vec{v} = \langle 1,-2

angle$, $\vec{w} = \langle 3,1
angle$

Step1: Identify base vectors

$\vec{v} = \langle 1,-2
angle$, $\vec{w} = \langle 3,1
angle$

Step2: Compute scalar multiples

$3\vec{v} = 3\langle 1,-2
angle = \langle 3,-6
angle$
$2\vec{w} = 2\langle 3,1
angle = \langle 6,2
angle$

Step3: Compute vector sum

$\vec{v} + \vec{w} = \langle 1+3, -2+1
angle = \langle 4,-1
angle$

Step4: Compute vector difference

$\vec{v} - \vec{w} = \langle 1-3, -2-1
angle = \langle -2,-3
angle$

Answer:

Part 1:

• $\vec{v} = \langle 1,0

angle$, $3\vec{v} = \langle 3,0
angle$

• $\vec{w} = \langle 0,1

angle$, $2\vec{w} = \langle 0,2
angle$

• $\vec{v} + \vec{w} = \langle 1,1

angle$, $\vec{v} - \vec{w} = \langle 1,-1
angle$

Part 2:

• $\vec{v} = \langle -1,1

angle$, $3\vec{v} = \langle -3,3
angle$

• $\vec{w} = \langle 1,2

angle$, $2\vec{w} = \langle 2,4
angle$

• $\vec{v} + \vec{w} = \langle 0,3

angle$, $\vec{v} - \vec{w} = \langle -2,-1
angle$

Part 3:

• $\vec{v} = \langle 1,-2

angle$, $3\vec{v} = \langle 3,-6
angle$

• $\vec{w} = \langle 3,1

angle$, $2\vec{w} = \langle 6,2
angle$

• $\vec{v} + \vec{w} = \langle 4,-1

angle$, $\vec{v} - \vec{w} = \langle -2,-3
angle$

(To sketch: Plot each vector as an arrow from the origin to its coordinate point on the Cartesian plane)