Question

assignment

part 1: complete the operations geometrically.

use vector operations to draw the resultant vector.

a) draw \\(2\mathbf{u} + 4\mathbf{v}\\). (3 points)

Explanation:

Identify the given vectors

We read the components of vectors \(\mathbf{u}\) and \(\mathbf{v}\) from the grid.
The initial point of both vectors is at the origin \((0,0)\).
The terminal point of \(\mathbf{u}\) (blue arrow) is at \((-1, 2)\).
Thus, \(\mathbf{u} = \langle -1, 2
angle\).
The terminal point of \(\mathbf{v}\) (orange arrow) is at \((1, 2)\).
Thus, \(\mathbf{v} = \langle 1, 2
angle\).

Calculate the scaled vectors

We multiply each vector by its respective scalar.

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

$$ 4\mathbf{v} = 4\langle 1, 2 angle = \langle 4, 8 angle $$

Find the resultant vector

We add the two scaled vectors together.

$$ 2\mathbf{u} + 4\mathbf{v} = \langle -2, 4 angle + \langle 4, 8 angle = \langle -2 + 4, 4 + 8 angle = \langle 2, 12 angle $$

Describe the geometric construction

To draw the resultant vector geometrically:

1. Draw \(2\mathbf{u}\) starting from the origin \((0,0)\) to \((-2, 4)\).
2. From the tip of \(2\mathbf{u}\) at \((-2, 4)\), draw \(4\mathbf{v}\) moving \(4\) units right and \(8\) units up, ending at \((2, 12)\).
3. Draw the resultant vector from the origin \((0,0)\) directly to the final point \((2, 12)\).

Answer:

To geometrically draw the resultant vector \(2\mathbf{u} + 4\mathbf{v}\):

1. Identify the components of the given vectors:

• \(\mathbf{u} = \langle -1, 2

angle\)

• \(\mathbf{v} = \langle 1, 2

angle\)

1. Calculate the scaled vectors:

• \(2\mathbf{u} = \langle -2, 4

angle\)

• \(4\mathbf{v} = \langle 4, 8

angle\)

1. Calculate the resultant vector:

• \(2\mathbf{u} + 4\mathbf{v} = \langle 2, 12

angle\)

1. Geometric Drawing Steps:

• Draw the vector \(2\mathbf{u}\) starting from the origin \((0,0)\) to the point \((-2, 4)\).
• From the terminal point \((-2, 4)\), draw the vector \(4\mathbf{v}\) (which goes \(4\) units to the right and \(8\) units up) to reach the terminal point \((2, 12)\).
• Draw the final resultant vector arrow starting from the origin \((0,0)\) and pointing directly to \((2, 12)\).