Question

in this activity, you will find vector sums and scalar multiples of vectors by graphing the vectors.
part a
vector u = <2, -5>. describe how you would use a graphical method to add vector u to vector v (shown in the graph). then, write the resultant vector in component form.
graph of a coordinate plane with vector v starting at the origin and ending at (3,1)

Explanation:

Step1: Determine vector v's components

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

Step2: Recall vector addition rule

To add vectors \( \mathbf{u} = \langle u_1, u_2
angle \) and \( \mathbf{v} = \langle v_1, v_2
angle \), we use \( \mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2
angle \).

Step3: Add the vectors

Given \( \mathbf{u} = \langle 2, -5
angle \) and \( \mathbf{v} = \langle 3, 1
angle \), we calculate:
\( u_1 + v_1 = 2 + 3 = 5 \)
\( u_2 + v_2 = -5 + 1 = -4 \)
So the resultant vector is \( \langle 5, -4
angle \).

(Graphical method: Draw \( \mathbf{v} \) from origin to \((3,1)\). Then draw \( \mathbf{u} \) starting at the tip of \( \mathbf{v} \) (so from \((3,1)\), move 2 units right and 5 units down to \((5, -4)\)). The resultant is from origin to \((5, -4)\).)

Answer:

The resultant vector in component form is \( \langle 5, -4
angle \).