Math Problem Statement

Review the graph.

On a coordinate plane, vector u has origin (negative 2, 3) and terminal point (6, 3).

What are the characteristics of the vector shown?

magnitude of 8 and direction angle equal to 0° magnitude of 8 and direction angle equal to 180° magnitude of 2 StartRoot 2 EndRoot and direction angle equal to 0° magnitude of 2 StartRoot 2 EndRoot and direction angle equal to 180°

Solution

To analyze the vector u\vec{u}, we’ll calculate its characteristics using its origin and terminal points.

Step 1: Determine the Vector Components

The vector u\vec{u} has an origin at (2,3)(-2, 3) and a terminal point at (6,3)(6, 3). The components of u\vec{u} can be calculated as: u=(x2x1,y2y1)=(6(2),33)=(8,0)\vec{u} = (x_2 - x_1, y_2 - y_1) = (6 - (-2), 3 - 3) = (8, 0)

Step 2: Calculate the Magnitude

The magnitude of u\vec{u} is given by: u=(8)2+(0)2=64=8|\vec{u}| = \sqrt{(8)^2 + (0)^2} = \sqrt{64} = 8

Step 3: Determine the Direction Angle

Since the vector lies on the x-axis and moves to the right (positive x-direction), its direction angle is 00^\circ.

Conclusion

The vector u\vec{u} has:

  • Magnitude of 8
  • Direction angle of 00^\circ

Answer

The correct option is: Magnitude of 8 and direction angle equal to 00^\circ

Would you like further clarification on any steps?

Here are some related questions to extend understanding:

  1. How would the vector’s direction angle change if it pointed left instead of right?
  2. What would the magnitude be if the vector spanned from (6, 3) to (-2, 3)?
  3. How can we find a unit vector with the same direction as u\vec{u}?
  4. How does the magnitude change if the vector's components double?
  5. What would the direction angle be for a vector lying on the y-axis?

Tip: A vector lying entirely on the x-axis has a direction angle of 00^\circ if it points to the right and 180180^\circ if it points to the left.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Vectors
Coordinate Geometry
Magnitude and Direction of a Vector

Formulas

Vector components formula: (x2 - x1, y2 - y1)
Magnitude formula: |u| = √((x2 - x1)^2 + (y2 - y1)^2)
Direction angle for horizontal vectors

Theorems

-

Suitable Grade Level

Grades 9-10