Question

Find the direction angle of \(\mathbf{v}\). \(\mathbf{v}=6 \mathbf{i}-4 \mathbf{j}\)

Short answer

The direction angle of \( \mathbf{v} = 6 \mathbf{i} - 4 \mathbf{j} \) is approximately \ 326.31°\.

Step-by-step solution

Identify Components

Recognize the components of the vector \(\mathbf{v}=6 \mathbf{i}-4 \mathbf{j}\) where its components are \( \mathbf{v}_x = 6 \) and \( \mathbf{v}_y = -4 \).

Understand the Direction Angle Formula

The direction angle \(\theta\) of the vector \(\mathbf{v} \) can be found using the formula \[ \theta = \tan^{-1} \left(\frac{ \mathbf{v}_y }{ \mathbf{v}_x }\right)\].

Substitute the Components

Substitute the values of the components into the directional angle formula: \[ \theta = \tan^{-1} \left(\frac{-4}{6} \right)\]. This simplifies to \[ \theta = \tan^{-1} \left(\frac{-2}{3} \right)\].

Compute the Inverse Tangent

Use a calculator to find \(\tan^{-1}\left(\frac{-2}{3}\right)\). This gives \[ \theta \approx -33.69°\].

Interpret the Result

Since the angle is negative and vectors in these problems are typically measured counterclockwise from the positive x-axis, add \( 360° \) to the result to convert it to a standard direction angle. \[ \theta = -33.69° + 360° = 326.31° \]

Key concepts

Vector Components

In this exercise, the vector \( \textbf{v} = 6\textbf{i} - 4\textbf{j} \) is given. To solve for the direction angle, we first break the vector into its components. A vector is typically expressed in terms of its components along the x-axis and y-axis.
\[ \textbf{v} = v_x \textbf{i} + v_y \textbf{j} \]
Where:

• \textbf{i} is the unit vector in the x-direction.
• \textbf{j} is the unit vector in the y-direction.

For our vector, the components are:
\textbf{v_x} = 6
\textbf{v_y} = -4
Understanding vector components helps us to use these values in trigonometric formulas to find the direction angle.

Inverse Tangent

To find the direction angle \( \theta \) of a vector from its components, we use the inverse tangent function (also called the arctangent). The formula for \( \theta \) is:
\[ \theta = \tan^{-1} \left( \frac{ \textbf{v_y} }{ \textbf{v_x} } \right) \]
The inverse tangent function, denoted as \( \tan^{-1} \) or \( \text{arctan} \), gives us the angle whose tangent is the given number. To apply this formula, we substitute our components:
\[ \theta = \tan^{-1} \left( \frac{ -4 }{ 6 } \right) = \tan^{-1} \left( \frac{ -2 }{ 3 } \right) \]
By using a calculator:
\[ \theta \approx -33.69^\text{°} \]
It’s important to use the calculator correctly, ensuring it’s in degree mode if angles are to be found in degrees.

Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between the angles and sides of triangles, especially right triangles. For vectors, trigonometry lets us convert between a vector’s components and its direction angle using trigonometric functions like sine, cosine, and tangent.
To interpret the angle from \( \theta = -33.69^\text{°} \):

• The angle is negative because it’s measured clockwise from the positive x-axis.
• For standard requirements where angles are measured counterclockwise, add \( 360° \) to the negative angle.

Thus:
\theta = -33.69^\text{°} + 360^\text{°} = 326.31^\text{°}'
This transformed angle indicates the direction of the vector measured counterclockwise from the positive x-axis. Understanding these conversions is crucial for accurately interpreting and using vector directions in various math and physics problems.