Problem 34 - Contemporary Precalculus

Question

Find the component form of the vector \(v\) whose magnitude and direction angle \(\theta\) are given. $$\|\mathbf{v}\|=3, \boldsymbol{\theta}=310^{\circ}$$

Step-by-Step Solution

Answer

Question: Find the component form of a vector with given magnitude and direction angle: magnitude = 3, angle = 310°. Answer: The component form of the vector is approximately ⟨1.634, -2.524⟩.

Step 1: Convert the angle to radians

To find the x and y components, we need the angle in radians. To convert an angle from degrees to radians, we use the following formula: Radians = Degrees × (π/180) So, for this problem, the angle in radians is: $$\boldsymbol{\theta}_{rad} = 310^{\circ}\times\left(\frac{\pi}{180}\right)$$

Step 2: Calculate the x and y components

Now we will use the formulas for the x and y components mentioned in the analysis to find the components of the vector: $$x = \|v\| \cdot \cos(\boldsymbol{\theta}_{rad})$$ $$y = \|v\| \cdot \sin(\boldsymbol{\theta}_{rad})$$ Plugging in the values of the magnitude and the angle in radians, we get: $$x = 3 \cdot \cos\left(310^{\circ}\times\left(\frac{\pi}{180}\right)\right)$$ $$y = 3 \cdot \sin\left(310^{\circ}\times\left(\frac{\pi}{180}\right)\right)$$

Step 3: Evaluate the expressions for x and y components

Now, we will use a calculator or a computer program to evaluate the expressions for the x and y components: $$x \approx 1.634$$ $$y \approx -2.524$$

Step 4: Write the vector in component form

Since we now have the x and y components, we can write the vector \(\mathbf{v}\) in component form: $$\mathbf{v} \approx \langle 1.634, -2.524 \rangle$$

Key Concepts

Magnitude and Direction Angle

When dealing with vectors, understanding their magnitude and direction angle is crucial. The magnitude, often represented as \(\|\mathbf{v}\|\), tells us how long or strong a vector is. It's like the vector's "size." Meanwhile, the direction angle \(\theta\) indicates the angle at which the vector is oriented relative to a reference direction, typically the positive x-axis.

In our exercise, the given magnitude is 3, meaning the vector stretches out to a length of 3 units. The direction angle is 310°, showing the angle between the vector and the positive x-axis, measured counterclockwise. Knowing these allows us to find the vector's components, essentially breaking it down into its parts along the x and y axes.

Degrees to Radians Conversion

Angles can be expressed in degrees or radians. Most calculators and mathematical functions use radians, which makes it important to convert degrees to radians for calculations.

To convert degrees to radians, use the formula:

• Radians = Degrees × \(\left(\frac{\pi}{180}\right)\)

For our problem where \(\theta = 310°\), the conversion to radians is
\(310° \times \left(\frac{\pi}{180}\right)\).
This conversion allows us to use trigonometric functions effectively, as they typically operate in radians.

Trigonometric Functions

Trigonometric functions like cosine (cos) and sine (sin) help us decompose vectors into their x and y components. Here's how it works:

• The x-component of a vector: \(x = \|v\| \cdot \cos(\boldsymbol{\theta}_{rad})\)
• The y-component of a vector: \(y = \|v\| \cdot \sin(\boldsymbol{\theta}_{rad})\)

By substituting the magnitude and the angle in radians, we find:
\(x = 3 \cdot \cos(310° \times \left(\frac{\pi}{180}\right))\)
\(y = 3 \cdot \sin(310° \times \left(\frac{\pi}{180}\right))\)

These formulas make it possible to break the vector into horizontal and vertical parts, revealing its true direction and size in a coordinate system.

Vector Representation in Precalculus

In precalculus, vectors are often represented in component form. This means expressing a vector as \(\mathbf{v} = \langle x, y \rangle\), where \(x\) and \(y\) are the vector's horizontal and vertical components respectively.

With our calculated components:

• \(x \approx 1.634\)
• \(y \approx -2.524\)

The vector \(\mathbf{v}\) can be written as \(\langle 1.634, -2.524 \rangle\).

This form is especially useful because it clearly indicates the vector's position in a two-dimensional space, helping visualize how it moves or points in a plane.