Question

b) draw \\(2\boldsymbol{u} - \frac{1}{3}\boldsymbol{v}\\). (4 points)

c) draw \\(\boldsymbol{u} + \boldsymbol{v} + \boldsymbol{w}\\). (4 points)

Explanation:

Identify vector components from the grids

Identify components of each vector.
For part b):
Vector \(\vec{u}\) starts at \((0,0)\) and ends at \((1,-2)\). Thus, \(\vec{u} = \langle 1, -2
angle\).
Vector \(\vec{v}\) starts at \((0,0)\) and ends at \((-6,-6)\). Thus, \(\vec{v} = \langle -6, -6
angle\).

For part c):
Vector \(\vec{u}\) starts at \((0,0)\) and ends at \((1,5)\). Thus, \(\vec{u} = \langle 1, 5
angle\).
Vector \(\vec{v}\) starts at \((0,0)\) and ends at \((-3,1)\). Thus, \(\vec{v} = \langle -3, 1
angle\).
Vector \(\vec{w}\) starts at \((0,0)\) and ends at \((4,-3)\). Thus, \(\vec{w} = \langle 4, -3
angle\).

Calculate the linear combination for part b)

Using Scalar Multiplication and Vector Addition:

$$ 2\vec{u} = 2\langle 1, -2 angle = \langle 2, -4 angle $$

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

Combine these components:

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

The resulting vector starts at \((0,0)\) and ends at \((4,-2)\).

Calculate the linear combination for part c)

Using Vector Addition for three Geometric Vectors:

$$ \vec{u} + \vec{v} + \vec{w} = \langle 1, 5 angle + \langle -3, 1 angle + \langle 4, -3 angle $$

Sum the horizontal components:

$$ x = 1 + (-3) + 4 = 2 $$

Sum the vertical components:

$$ y = 5 + 1 + (-3) = 3 $$

The resulting vector starts at \((0,0)\) and ends at \((2,3)\).

Answer:

Question b)

The vector \(2\vec{u} - \frac{1}{3}\vec{v}\) is represented by an arrow starting at the origin \((0,0)\) and ending at the coordinate point \((4, -2)\).

Question c)

The vector \(\vec{u} + \vec{v} + \vec{w}\) is represented by an arrow starting at the origin \((0,0)\) and ending at the coordinate point \((2, 3)\).