In Problems 65-72, find the direction angle of 𝐯. 𝐯=-𝐢-5 𝐣 |...

Key Concepts

Direction Angle

The direction angle of a vector is the angle measured from the positive x-axis to the vector. It indicates the orientation of the vector in the plane and is typically measured in degrees or radians. When working with two-dimensional vectors, this concept helps in visualizing and resolving the vector into its directional components.

Two-Dimensional Vector Components

Every vector in the plane can be broken down into its x (horizontal) and y (vertical) components. These components are used to determine the overall magnitude and direction of the vector. Understanding how to work with these components is essential for calculating things such as the vector's direction angle.

Inverse Tangent Function

The inverse tangent function, or arctan, is used to calculate the angle that a vector makes with the horizontal axis by taking the ratio of its vertical component to its horizontal component. However, since arctan returns values typically in a restricted range, additional considerations are needed to determine the correct quadrant for the angle.

Quadrant Analysis

When using inverse trigonometric functions to determine direction angles, it is important to consider which quadrant the vector lies in. This is because the simple output from an arctan calculation may not reflect the true direction if the vector components indicate that it is in a quadrant other than the first. Adjustments must be made to the basic arctan output to correctly represent the angle based on the signs of the vector components.

In Problems 65-72, find the direction angle of $\mathbf{v}$. $\mathbf{v}=-\mathbf{i}-5 \mathbf{j}$

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