Question

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

Short answer

The direction angle of \(\textbf{v} = \textbf{\textbf{v}}\textbf{y} + \textbf{v_y}\).\theta = 60^\text{}.

Step-by-step solution

Understand the Vector Components

Consider the given vector \(\textbf{v} = \textbf{i} + \sqrt{3} \textbf{j}\). Here, the vector \(\textbf{v}\) has an x-component of 1 (the coefficient of \(\textbf{i}\)) and a y-component of \(\textbf{j}\), which is \(\textbf{v_y} = \sqrt{3}\).

Recall the Formula for the Direction Angle

The direction angle \(\theta\) of the vector \(\textbf{v}\) can be found using the formula \(\theta = \text{arctan}\frac{{v_y}}{{v_x}}\).

Substitute the Components into the Formula

Substitute the x-component and y-component into the formula: \(\theta = \text{arctan}\frac{\frac{\tf{\textbf{v_y = \sqrt{3}}}}1}\).

Calculate the Value

Calculate the arctangent of the fraction: \(\theta = \text{arctan}\frac{\textbf{v_y}}{\textbf{v_x}} = \text{arctan}\frac{{\text{\frac{\textbf{1}}}} = 60\^\text{}\).

Convert to Degrees if Necessary

The calculated direction angle \(\theta\) is in degrees. If radians are preferred, convert accordingly (1 degree = \frac{\textbf{\text{\frac{\pi}}\textbf{180}}}). In this case, the direction angle is already in degrees.

Key concepts

Vector Components

When working with vectors, it's essential to understand their components. A vector, like \(\textbf{v} = \textbf{i} + \sqrt{3} \textbf{j}\), has an x-component and a y-component. The x-component corresponds to the horizontal direction, and the y-component corresponds to the vertical direction.

For our vector \(\textbf{v}\), the x-component is 1 (since it is the coefficient of \(\textbf{i}\)), and the y-component is \(\textbf{\textbf{v}_y} = \sqrt{3}\). These components can be visualized as forming a right triangle, with the vector itself representing the hypotenuse. Understanding the components helps us in performing calculations like finding the direction angle.

Arctangent Function

The arctangent function is vital in determining the direction angle of a vector. It is the inverse of the tangent function. If you know the opposite side (y-component) and adjacent side (x-component) of a right triangle, you can find the angle \(\theta\) using arctangent.

The arctangent function is denoted as \(\text{arctan}\) or sometimes as \(\tan^{-1}\). For our case, if \(\theta\) is the angle, then:

\(\theta = \text{arctan} \frac{y}{x}\).

This function will give you the angle in radians by default, but you can convert it to degrees if needed (multiply by \(\frac{180}{\text{\textbf{\text{\textpi}}}}\)).

Direction Angle Calculation

To calculate the direction angle of a vector, you need both the x and y components. For our vector \(\textbf{v} = \textbf{i} + \sqrt{3} \textbf{j}\), we have \(\textbf{\textbf{v}_x} = 1\) and \(\textbf{\textbf{v}_y} = \sqrt{3}\).

Plug these values into the formula for the direction angle:

\(\theta = \text{arctan} \frac{\textbf{\textbf{v}_y}}{\textbf{\textbf{v}_x}}\).

Substituting the components, we get:

\(\theta = \text{arctan} \frac{\textbf{\textbf{v}_y} = \sqrt{3}}{1}\)

Now, \(\theta = \text{arctan}( \sqrt{3} )\) which simplifies to \(\theta = 60^\text{\textdegree} \) (since \(\text{arctan}( \sqrt{3} )\) equals 60 degrees). This gives us the direction angle of the vector.

Unit Vectors

Unit vectors are vectors with a magnitude of 1. They are used to specify directions in space. Common unit vectors include \(\textbf{\textbf{i}}\) and \(\textbf{\textbf{j}}\), which represent the x and y axes, respectively.

Any vector can be expressed as the sum of its components along these unit vectors. For instance, \(\textbf{v} = \textbf{\textbf{i}} + \sqrt{3} \textbf{\textbf{j}}\) shows how our vector \(\textbf{v}\) is built from the unit vectors \(\textbf{\textbf{i}}\) and \(\textbf{\textbf{j}}\).

Understanding unit vectors is crucial when decomposing vectors into components or when performing any vector calculations.