Question

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

Explanation:

Identify vectors

Identify components from graph.

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

$$ \boldsymbol{v} = \langle -6, -6 angle $$

Scale vectors

Apply Scalar Multiplication to each.

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

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

Combine components

Apply Vector Addition to find result.

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

Verify coordinates

Confirm the final vector.
The vector starts at \((0,0)\) and ends at \((4, -2)\).

Answer:

The vector \(2\boldsymbol{u} - \frac{1}{3}\boldsymbol{v}\) is represented in component form as:

$$ \langle 4, -2 angle $$

To draw this vector on the grid, start an arrow at the origin \((0,0)\) and place its terminal point (tip) at the coordinate \((4, -2)\).