Question

part b
vector ( mathbf{v} = langle 3, 1
angle ). describe how you would use a graphical method to add vector ( mathbf{v} ) to vector ( mathbf{u} ) (shown in the graph below). then, write the resultant vector in component form.
graph of a coordinate plane with vector ( mathbf{u} ) drawn from the origin to a point, and a grid with x-axis from -5 to 5 and y-axis from -5 to 5

Explanation:

Step1: Determine vector \(\mathbf{u}\)'s components

From the graph, vector \(\mathbf{u}\) starts at the origin \((0,0)\) and ends at \((3, -5)\), so \(\mathbf{u} = \langle 3, -5
angle\). Vector \(\mathbf{v} = \langle 1, 1
angle\) (given).

Step2: Graphical addition (Triangle/Parallelogram method)

1. Plot \(\mathbf{u}\): Draw vector \(\mathbf{u}\) from \((0,0)\) to \((3, -5)\).
2. Plot \(\mathbf{v}\) from \(\mathbf{u}\)'s tip: Place the tail of \(\mathbf{v}\) at \((3, -5)\), then draw \(\mathbf{v}\) (1 unit right, 1 unit up) to \((3 + 1, -5 + 1) = (4, -4)\).
3. Resultant vector \(\mathbf{u} + \mathbf{v}\): Draw from \((0,0)\) to \((4, -4)\) (or use component addition: \(\langle 3 + 1, -5 + 1

angle\)).

Step3: Component - form addition

For vectors \(\mathbf{u} = \langle u_x, u_y
angle\) and \(\mathbf{v} = \langle v_x, v_y
angle\), the resultant is \(\langle u_x + v_x, u_y + v_y
angle\).
Substitute \(u_x = 3\), \(u_y = -5\), \(v_x = 1\), \(v_y = 1\):
\(\mathbf{u} + \mathbf{v} = \langle 3 + 1, -5 + 1
angle = \langle 4, -4
angle\).

Answer:

The resultant vector in component form is \(\boldsymbol{\langle 4, -4
angle}\).
(Graphical method: Plot \(\mathbf{u}\), then plot \(\mathbf{v}\) from \(\mathbf{u}\)'s tip; the resultant connects the origin to \(\mathbf{v}\)'s tip.)