Problem 29 - College Trigonometry

Question

Approximate the component form of the vector \(\vec{v}\) using the information given about its magnitude and direction. Round your approximations to two decimal places. \(\|\vec{v}\|=450\); when drawn in standard position \(\vec{v}\) makes a \(210.75^{\circ}\) angle with the positive \(x\) -axis

Step-by-Step Solution

Answer

The vector \( \vec{v} \) is approximately \( \langle -389.80, -242.70 \rangle \).

Step 1: Convert angle to radians

To perform calculations with angles in trigonometric functions, it's better to first convert the angle from degrees to radians. The conversion is done by multiplying the degree measure by \( \frac{\pi}{180} \). Thus, \( \theta = 210.75^{\circ} \) is equivalent to:\[ \theta = 210.75 \times \frac{\pi}{180} \approx 3.677274 \text{ radians} \]

Step 2: Calculate the x-component

The x-component of the vector \( \vec{v} \) can be found using the formula \( v_x = \|\vec{v}\| \cdot \cos(\theta) \). Using the given magnitude and the angle in radians, calculate as follows:\[ v_x = 450 \cdot \cos(3.677274) \approx -389.80 \]

Step 3: Calculate the y-component

Similarly, the y-component of the vector \( \vec{v} \) is calculated using \( v_y = \|\vec{v}\| \cdot \sin(\theta) \). Thus:\[ v_y = 450 \cdot \sin(3.677274) \approx -242.70 \]

Step 4: State the vector in component form

The vector \( \vec{v} \) can be written using its x and y components in component form. Thus, \( \vec{v} \approx \langle -389.80, -242.70 \rangle \).

Key Concepts

Vector Magnitude

Vector magnitude represents the length of the vector and acts as a scalar quantity, meaning it doesn't have any direction associated with it. It's denoted by \( \|\vec{v}\| \) and is essential for finding the vector's components.

In this exercise, the given magnitude is 450. This means the vector stretches over a length of 450 units in space, regardless of its direction—just like the size of an arrow representing the vector.

The scalar value of a vector is particularly important because it's used in conjunction with its direction to derive its components. Think of the magnitude like knowing the speed in a specific direction, helping us understand how far the vector reaches. Calculating the magnitude involves taking the square root of the sum of the squares of its components—though in this case, we’ve been directly given this value.

Angle Conversion to Radians

Angles can be measured in degrees or radians, and converting between these two units is crucial, especially when dealing with trigonometric functions in mathematics.

The standard practice in math and physics is to perform calculations in radians because they relate more directly to arc lengths in circles. To convert degrees to radians, multiply the degree value by \( \frac{\pi}{180} \).

For example, the angle \( 210.75^{\circ} \) can be converted into radians by the formula:

• \( \theta = 210.75 \times \frac{\pi}{180} \approx 3.677274 \text{ radians} \)

This conversion is necessary for the next steps because trigonometric functions like cosine and sine require input in radians for accuracy when solving problems involving vectors.

Trigonometric Components

When a vector is presented in standard position, its direction forms an angle with the positive x-axis. To determine the vector's components along the x and y axes, we use trigonometric functions—cosine and sine, respectively.

Calculating the x-component: The x-component \( v_x \) of a vector can be isolated using the cosine of its angle:

• \( v_x = \|\vec{v}\| \cdot \cos(\theta) \)

In this exercise, with \( \|\vec{v}\| = 450 \) and \( \theta \approx 3.677274 \) radians, we find:

• \( v_x = 450 \cdot \cos(3.677274) \approx -389.80 \)

Calculating the y-component: Similarly, the y-component \( v_y \) utilizes the sine of the angle:

• \( v_y = \|\vec{v}\| \cdot \sin(\theta) \)

With the same magnitude and angle, calculate:

• \( v_y = 450 \cdot \sin(3.677274) \approx -242.70 \)

These calculations give us the vector in component form: \( \vec{v} \approx \langle -389.80, -242.70 \rangle \). This form illustrates how much of the vector's total magnitude stretches along the x and y axes, providing a comprehensive view of its direction and length in the plane.