Question

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

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

Explanation:

Step1: Define given 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

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

Step5: Describe sketching rules

• $\vec{v}$: From origin to $(1,0)$
• $3\vec{v}$: From origin to $(3,0)$
• $\vec{w}$: From origin to $(0,1)$
• $2\vec{w}$: From origin to $(0,2)$
• $\vec{v}+\vec{w}$: From origin to $(1,1)$
• $\vec{v}-\vec{w}$: From origin to $(1,-1)$

Answer:

• $\vec{v}$: Vector from $(0,0)$ to $(1,0)$
• $3\vec{v}$: Vector from $(0,0)$ to $(3,0)$
• $\vec{w}$: Vector from $(0,0)$ to $(0,1)$
• $2\vec{w}$: Vector from $(0,0)$ to $(0,2)$
• $\vec{v}+\vec{w}$: Vector from $(0,0)$ to $(1,1)$
• $\vec{v}-\vec{w}$: Vector from $(0,0)$ to $(1,-1)$

(When sketching, plot each endpoint and draw an arrow from the origin to each point to represent the vector.)