Find the magnitude and direction angle (to the nearest...

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in $\left[0,360^{\circ}\right)$.
$$\langle 8 \sqrt{2},-8 \sqrt{2}\rangle$$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Vector Representation in Two Dimensions

A vector in the plane is typically expressed in component form, which displays its horizontal and vertical components. This form is useful for determining properties such as magnitude and direction using basic algebraic and trigonometric operations.

Vector Magnitude

The magnitude of a vector, also known as its Euclidean norm, represents the length of the vector. It is calculated using the Pythagorean theorem by taking the square root of the sum of the squares of its horizontal and vertical components.

Direction Angle of a Vector

The direction angle is the measure of the angle a vector makes with the positive x-axis. This angle is commonly found using the inverse tangent function (arctan) of the ratio between the vertical and horizontal components, providing the initial measure of the angle.

Quadrant Consideration and Angle Adjustment

Since the arctan function alone does not account for the sign of both components, it is important to consider the quadrant where the vector lies. Depending on the signs of the components, adjustments such as adding 180 degrees might be required to obtain the correct angle within a specified range, typically [0, 360°).

Transcript

Please give Ace some feedback