Problem 67 - Algebra and Trigonometry

Question

Find the direction angle of \(\mathbf{v}\). \(\mathbf{v}=3 \mathbf{i}+3 \mathbf{j}\)

Step-by-Step Solution

Answer

The direction angle is 45°.

Step 1: Identify the Vector Components

The vector \(\textbf{v} = 3 \mathbf{i} + 3 \mathbf{j}\) has components \(v_x = 3\) and \(v_y = 3\).

Step 2: Use the Formula for the Direction Angle

The direction angle \(\theta\) of a vector can be found using the formula \(\theta = \tan^{-1} \left( \frac{v_y}{v_x} \right)\).

Step 3: Calculate the Direction Angle

Substitute the values \(v_x = 3\) and \(v_y = 3\) into the formula: \(\theta = \tan^{-1} \left( \frac{3}{3} \right) = \tan^{-1} (1)\).

Step 4: Determine the Inverse Tangent

The value of \(\tan^{-1}(1)\) is \(\theta = 45^\text{°}\).

Key Concepts

Vector Components

In vector mathematics, a vector has both direction and magnitude. It is often represented in terms of its components along the coordinate axes. For a 2D vector, these components are typically denoted as \(v_x\) and \(v_y\).

The given vector \(\mathbf{v} = 3 \, \mathbf{i} + 3 \, \mathbf{j}\) means that the vector has a component of 3 units in the direction of the x-axis (represented by the unit vector \(\mathbf{i}\)) and a component of 3 units in the direction of the y-axis (represented by the unit vector \(\mathbf{j}\)).

To summarize:

• \(v_x = 3\)
• \(v_y = 3\)

Understanding these components is crucial for calculating other vector properties like magnitude and direction (angle).

Inverse Tangent

The inverse tangent function, denoted as \tan^{-1}\ or arctangent, is used to determine the angle whose tangent is a given number. In our case, it helps find the direction angle \(\theta\) of the vector using its components.

The formula we use is: \theta = \tan^{-1} \left( \frac{v_y}{v_x} \right)\.

For the given vector components: \[ \theta = \tan^{-1} \left( \frac{3}{3} \right) = \tan^{-1} (1) \] The inverse tangent of 1 is a well-known value: \[ \theta = 45^\circ \] Remember, this angle is measured counterclockwise from the positive x-axis.

This effectively means that our vector points equally along both the x and y directions.

Trigonometry

Trigonometry is the branch of mathematics dealing with relationships between the angles and sides of triangles. It is also extensively used in vector mathematics for problems like this one.

Here, we are leveraging the tangent function to find the direction angle of a vector. Tangent, in a right-angled triangle, is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \] In the context of vector components, 'opposite' corresponds to \(v_y\) and 'adjacent' corresponds to \(v_x\).

Therefore, the tangent function can help us find the angle \(\theta\) that the vector makes with the x-axis.

To find the angle, we use the inverse tangent function as explained: \[ \theta = \tan^{-1} \left( \frac{v_y}{v_x} \right) \] This is a practical application of trigonometry in vector analysis, giving us crucial information about the vector's direction.