Question

In Problems 65-72, find the direction angle of $\mathbf{v}$. $\mathbf{v}=6 \mathbf{i}-4 \mathbf{j}$

   In Problems 65-72, find the direction angle of $\mathbf{v}$.
$\mathbf{v}=6 \mathbf{i}-4 \mathbf{j}$

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 70

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For the vector $\mathbf{v} = 6\mathbf{i} - 4\mathbf{j}$, the components are $x = 6$ and $y = -4$. Show more…

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In Problems 65-72, find the direction angle of $\mathbf{v}$. $\mathbf{v}=6 \mathbf{i}-4 \mathbf{j}$

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Key Concepts

Trigonometric Functions (Arctan)

The arctan (or inverse tangent) function is used to find the angle whose tangent is the ratio of the y-component to the x-component of a vector. This function provides an initial angle measurement which might need further adjustment to accurately represent the vector's direction.

Quadrant Analysis

When determining the direction angle, it is crucial to consider the signs of the vector components to identify the correct quadrant in the coordinate plane. Since the arctan function typically only returns values in a limited range, additional adjustments or conditions must be applied to obtain the true direction angle relative to the positive x-axis.

Direction Angle

The direction angle of a vector is the angle that the vector makes with the positive x-axis. It is an important concept in vector analysis that represents the orientation of the vector relative to a reference direction in the coordinate system.

Vector Components

Vectors in the plane can be decomposed into horizontal (x) and vertical (y) components. Understanding these components is fundamental because the computation of the direction angle is based on the ratio of the vertical component to the horizontal component.

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