Consider the vectors Bold uuequals=left angle 4 comma 0 right angle4,0 and Bold vvequals=left angle negative 3 comma negative 3 right angle−3,−3. Sketch the vectors, find the angle between the vectors, and compute the dot product using the definition Bold u times Bold v equals StartAbsoluteValue Bold u EndAbsoluteValue StartAbsoluteValue Bold v EndAbsoluteValue cosine thetau•v=u vcosθ.

Angle between the two vectors is 135°

u•v = -12

Step-by-step explanation:

Given two vectors u = (4,0) and v = (-3,-3).

To find the angle between the two vectors we will use the formula for calculating the angle between two vectors as shown;

u•v = |u||v|cos theta

cos theta = u•v/|u||v|

theta = arccos  (u•v/|u||v|)

u•v = (4,0)•(-3,-3)

u•v = 4(-3)+0(-3)

u•v = -12

For |u| and |v|

|u| = √4²+0²

|u| = √16 = 4

|v| = √(-3)²+(-3)²

|v| = √9+9

|v| = √18

|v| = 3√2

|u||v| = 4×3√2 = 12√2

theta = arccos(-12/12√2)

theta = arccos(- 1/√2)

theta = -45°

Since cos is negative in the second quadrant, theta = 180-45°

theta = 135°

To get u•v using the formula u•v = |u||v|cos theta

Given |u||v| = 12√2 and theta = 135°

u•v = 12√2cos 135°

u•v = 12√2× -1/√2

u•v = -12√2/√2

u•v = -12

For the diagram of the vectors, find it in the attachment below.