Problem 69 - Precalculus with Limits

Question

Finding the Component Form of a Vector In Exercises \(67-74\) , find the component form of \(v\) given its magnitude and the angle it makes with the positive \(x\) -axis. Sketch y. $$\begin{array}{ll}{\text { Magnitude }} & {\text { Angle }} \\\ {\|\mathbf{v}\|=\frac{7}{2}} & {\theta=150^{\circ}}\end{array}$$

Step-by-Step Solution

Answer

The component form of the vector \( v \) after carrying out the steps will be your answer.

Step 1: Conversion of Angle

First, convert the given angle from degrees to radians as trigonometric computations in calculus usually presume radians as the unit for angles. The conversion factor to move from degrees to radians is \( \frac{\pi}{180} \). So, \( \theta =150^{\circ}=150 * \frac{\pi}{180} = \frac{5\pi}{6} \) rad.

Step 2: Compute the x-component

The x-component of the vector can be found using the magnitude of the vector and the cosine of the angle it makes with the positive x-axis. This is given by \( x = \|v\| * cos( \theta ) \). Use the given magnitude \( \|v\| = \frac{7}{2} \) and the converted angle. Therefore, \( x = \frac{7}{2} * cos( \frac{5\pi}{6} ) \). Compute this to obtain the x-component of the vector.

Step 3: Compute the y-component

The y-component of the vector can be found using the magnitude of the vector and the sine of the angle it makes with the positive x-axis. This is given by \( y = \|v\| * sin( \theta ) \). Use the given magnitude \( \|v\| = \frac{7}{2} \) and the converted angle. Therefore, \( y = \frac{7}{2} * sin( \frac{5\pi}{6} ) \). Compute this to obtain the y-component of the vector.

Step 4: Component Form of the Vector

The component form of vector \( v \) is given by \( v = \). Substitute the obtained x and y components into this to get the component form.

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are pivotal when dealing with angles and rotations in mathematics. They help calculate relationships within triangles. In particular, sine and cosine are two core functions:

• Sine (\( \sin \)): This represents the ratio of the opposite side to the hypotenuse in a right-angled triangle.
• Cosine (\( \cos \)): This is the ratio of the adjacent side to the hypotenuse.

These functions allow us to transition from a vector's angular view to a compact component form. The component form is essential in representing a vector in vector calculations such as in physics and engineering. A vector's x-component can be derived using the cosine of its angle, and the y-component is found using the sine.

Angle Conversion Made Easy

Angles are usually given in degrees, especially in geometric contexts, but calculus and physics often require these angles in radians. Converting degrees to radians is straightforward with the formula:- Multiply the degree value by \( \frac{\pi}{180} \).
This step changes a degree measure into radians, which are more conducive for mathematical analysis as they simplify integral calculus applications. For example, converting \( 150^{\circ} \) involves:\[ 150^{\circ} \times \frac{\pi}{180} = \frac{5\pi}{6} \text{ radians} \].
So, understanding this conversion is crucial for smooth transitions between different mathematical contexts and solving problems accurately.

Defining Vector Magnitude

The vector magnitude is essentially the vector's length. Think of a vector as an arrow; the magnitude is how long that arrow is. In mathematics, it is denoted as \( \|\mathbf{v}\| \). It provides key insight into the size of the vector without considering its direction.

• Calculation: Often calculated in relation to its components, using \( \sqrt{x^2 + y^2} \) for a 2D vector.
• Usage: In exercises, magnitude helps determine the vector's reach across dimensions, influencing both the x and y components in our earlier example. Given \( \|\mathbf{v}\| = \frac{7}{2} \) means this vector has a specific half-real-number length, regardless of its direction.

Recognizing how magnitude influences vector dimensional analysis allows seamless computation and understanding of vectors in various spaces.