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

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

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 66

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In this case, $\mathbf{v} = \mathbf{i} + \sqrt{3} \mathbf{j}$. This means the vector $\mathbf{v}$ has components $x = 1$ and $y = \sqrt{3}$. Show more…

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

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Key Concepts

Vector Representation in Cartesian Coordinates

Vectors in the plane are commonly expressed in terms of their x- and y-components using standard unit vectors. This approach allows for a clear depiction of both magnitude and direction. It forms the basis for many operations in analytic geometry and linear algebra, where operations are performed component-wise.

Direction Angle of a Vector

The direction angle of a vector is the angle measured from the positive x-axis to the vector, typically in a counterclockwise direction. This angle provides a concise description of the vector's orientation in the plane, and is a fundamental concept in both geometry and physics.

Inverse Trigonometric Functions

Inverse trigonometric functions, particularly the arctan (or inverse tangent) function, are used to calculate the angle a vector makes with the x-axis by taking the ratio of the y-component to the x-component. This mathematical tool is essential in converting Cartesian coordinates into polar form.

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