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In Problems 65-72, find the direction angle of $\mathbf{v}$. $\mathbf{v}=-\mathbf{i}+3 \mathbf{j}$

   In Problems 65-72, find the direction angle of $\mathbf{v}$.
 $\mathbf{v}=-\mathbf{i}+3 \mathbf{j}$

Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry

Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry

Michael Sullivan 4th Edition

Chapter 8, Problem 72

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The vector given is \(\mathbf{v} = -\mathbf{i} + 3\mathbf{j}\). In component form, this can be written as \(\mathbf{v} = (-1, 3)\), where \(-1\) is the x-component and \(3\) is the y-component of the vector. Show more…

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In Problems 65-72, find the direction angle of $\mathbf{v}$. $\mathbf{v}=-\mathbf{i}+3 \mathbf{j}$

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Key Concepts

Vector Representation

This concept involves describing a vector in terms of its components along the standard basis vectors (typically i, j, and k in three dimensions). Understanding how a vector is represented by its components is fundamental to analyzing its magnitude and direction.

Direction Angle

The direction angle of a vector is the angle measured from the positive x?axis to the vector, normally expressed in degrees or radians. This angle indicates the orientation of the vector in the coordinate plane.

Inverse Trigonometric Functions

In the process of finding a vector’s direction angle, the inverse tangent function (arctan) is typically used to calculate the angle formed by the ratio of the vector's vertical component to its horizontal component. This function provides the acute approach angle, which might need adjustment based on the vector’s quadrant.

Quadrant Consideration

Because the arctan function only returns angles in a specific range, it is important to analyze which quadrant the vector lies in to determine the proper direction angle. Adjustments may be required to convert the initial angle value into the appropriate angle that accurately represents the vector’s orientation in the coordinate system.

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