Question

d) draw \\(u - v + w\\). (4 points)

Explanation:

Identify components of given vectors

The image is rotated \(90^\circ\) clockwise. Let's read the coordinates from the grid by orienting ourselves with the axes:

• The horizontal axis is labeled \(y\) (pointing right, with positive values \(1, 2, 3, 4, 5, 6\)).
• The vertical axis is labeled \(x\) (pointing downwards, with positive values \(1, 2, 3, 4, 5\) going down, and negative values \(-1, -2, -3, -4, -5\) going up).
• Let's write the vectors in standard Cartesian coordinates \((y, x)\) where \(y\) is the horizontal component and \(x\) is the vertical component (pointing upwards as positive, so we invert the vertical axis values to standard Cartesian coordinates where up is positive):
• Vector \(\vec{u}\): starts at \((0,0)\), ends at horizontal \(-4\), vertical \(-3\) (which is \(3\) units up on the grid). Thus, \(\vec{u} = \langle -4, 3

angle\).

• Vector \(\vec{v}\): starts at \((0,0)\), ends at horizontal \(-1\), vertical \(-4\) (which is \(4\) units up on the grid). Thus, \(\vec{v} = \langle -1, 4

angle\).

• Vector \(\vec{w}\): starts at \((0,0)\), ends at horizontal \(3\), vertical \(3\) (which is \(3\) units down on the grid). Thus, \(\vec{w} = \langle 3, -3

angle\).

Calculate the resultant vector

We need to find the vector \(\vec{r} = \vec{u} - \vec{v} + \vec{w}\):

$$
LATEXBLOCK0
$$

Determine the coordinates for drawing

The resultant vector \(\vec{r} = \vec{u} - \vec{v} + \vec{w} = \langle 0, -4
angle\).

• In the grid's coordinate system:
• Horizontal component (\(y\)-axis) is \(0\).
• Vertical component (\(x\)-axis) is \(-4\) in standard Cartesian (which corresponds to \(4\) units down on the grid, labeled as \(4\) on the vertical axis).
• Therefore, the vector starts at the origin \((0,0)\) and ends at \((0, 4)\) on the grid (along the vertical axis labeled \(x\)).

Answer:

To find and draw the vector \(\vec{u} - \vec{v} + \vec{w}\), we first determine the components of each vector from the grid:

• \(\vec{u} = \langle -4, 3

angle\)

• \(\vec{v} = \langle -1, 4

angle\)

• \(\vec{w} = \langle 3, -3

angle\)

Calculating the resultant vector:

$$ \vec{u} - \vec{v} + \vec{w} = \langle -4 - (-1) + 3, \; 3 - 4 - 3 angle = \langle 0, -4 angle $$

The final vector \(\vec{u} - \vec{v} + \vec{w}\) starts at the origin \((0,0)\) and ends at the coordinate \((0, -4)\) (which points \(4\) units straight down along the vertical axis).