Problem 70 - Algebra and Trigonometry

Question

Find the direction angle of \(\mathbf{v}\). \(\mathbf{v}=-5 \mathbf{i}-5 \mathbf{j}\)

Step-by-Step Solution

Answer

The direction angle of \(\textbf{v} = -5 \textbf{i} - 5 \textbf{j}\) is \(225^\text{\textdegree}\).

Step 1 - Identify Vector Components

The given vector is \(\textbf{v} = -5 \textbf{i} - 5 \textbf{j}\). Identify the components of the vector. Here, the x-component, \(v_x\), is -5 and the y-component, \(v_y\), is also -5.

Step 2 - Use the Formula

The formula to find the direction angle \( \theta \) of a vector \( \textbf{v} = v_x \textbf{i} + v_y \textbf{j} \) is \(\theta = \tan^{-1} \frac{v_y}{v_x} \).

Step 3 - Substitute the Values

Substitute the components \(-5\) for \(\textbf{i}\) and \(-5\) for \(\textbf{j}\) into the formula: \(\theta = \tan^{-1} \frac{-5}{-5} = \tan^{-1}(1)\).

Step 4 - Calculate the Angle

Calculate \( \tan^{-1}(1) \), which is \(45^\text{\textdegree}\).

Step 5 - Determine Correct Quadrant

Since both components are negative, the vector lies in the third quadrant. Therefore, the direction angle is \(180^\text{\textdegree} + 45^\text{\textdegree} = 225^\text{\textdegree}\).

Key Concepts

Vector Components

Vectors have both magnitude and direction. They can be broken down into components along the x-axis and y-axis. For the vector \(\mathbf{v} = -5 \mathbf{i} - 5 \mathbf{j}\), the components are:

• x-component: \(v_x = -5\)
• y-component: \(v_y = -5\)

Think of these components as describing how far and in which direction the vector goes along each axis. Knowing the components is crucial for further calculations, like finding the direction angle.

Tangent Inverse

To find the direction angle of a vector, you use the inverse tangent function, denoted as \(\tan^{-1}\).
The formula to find the angle \(\theta\) is:
\[ \theta = \tan^{-1} \frac{v_y}{v_x} \]
This function helps us determine the angle between the vector and the positive x-axis.
Make sure you have both components identified before you use the formula.

Quadrant Determination

Vectors can lie in any of the four quadrants of the Cartesian plane:

• First Quadrant: Both components are positive.
• Second Quadrant: x-component is negative, y-component is positive.
• Third Quadrant: Both components are negative.
• Fourth Quadrant: x-component is positive, y-component is negative.

In our example vector \(\mathbf{v} = -5 \mathbf{i} - 5 \mathbf{j}\), both components are negative. So, the vector is in the third quadrant. This information is essential for adjusting the direction angle correctly.

Direction Angle Calculation

Once you know the vector's components and quadrant, you can calculate the direction angle. Here's the process for our given vector \(\mathbf{v} = -5 \mathbf{i} - 5 \mathbf{j}\):

• First, plug the components into the tangent inverse formula: \[ \theta = \tan^{-1} \frac{-5}{-5} = \tan^{-1}(1) \]
  This gives \(\theta = 45^\text{\degree}\).
• Next, adjust the angle based on which quadrant the vector is in.
  Since \(\mathbf{v}\) is in the third quadrant, add \(180^\text{\degree}\) to the calculated angle: \[ 45^\text{\degree} + 180^\text{\degree} = 225^\text{\degree} \]

So, the direction angle of the vector is \(225^\text{\degree}\).
Remember, the adjustment step is crucial for getting the correct direction.