Problem 28 - Contemporary Precalculus

Question

Find the component form of the vector \(v\) whose magnitude and direction angle \(\theta\) are given. $$\|\mathbf{v}\|=5, \theta=30^{\circ}$$

Step-by-Step Solution

Answer

Answer: The component form of the vector is $\left<\frac{5\sqrt{3}}{2}, \frac{5}{2}\right>$.

Step 1: Convert direction angle to radians

First, we need to convert the direction angle from degrees to radians. To do this, we can use the formula: $$\text{radians} = \frac{\text{degrees}}{180} \cdot \pi$$ So, converting 30° to radians: $$\text{radians}= \frac{30}{180} \cdot \pi = \frac{\pi}{6} $$

Step 2: Find the x and y components of the vector

Next, we can find the x and y components of the vector using the magnitude and the direction angle in radians. The x and y components can be found using the cosine and sine functions: $$ x = \|v\|\cdot \cos{\theta} \\ y = \|v\|\cdot \sin{\theta} $$ Now we plug the values from the given exercise: $$ x = 5\cdot \cos\left(\frac{\pi}{6}\right) \\ y = 5\cdot \sin\left(\frac{\pi}{6}\right) $$ We know that \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\) and \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\), so we have: $$ x = 5 \cdot \frac{\sqrt{3}}{2} \\ y = 5 \cdot \frac{1}{2} $$

Step 3: Write the vector in component form

Finally, we can write the vector in component form using the x and y components we found in step 2: $$ \mathbf{v} = \left<\frac{5\sqrt{3}}{2}, \frac{5}{2}\right> $$ So, the component form of the vector is \(\left<\frac{5\sqrt{3}}{2}, \frac{5}{2}\right>\).

Key Concepts

Understanding Direction Angles

A direction angle represents the angle a vector forms with the positive x-axis. It provides information about the vector's orientation in a plane. For a vector given in polar coordinates with a direction angle \(\theta\), understanding this concept is key to converting the vector into its component form. If you imagine a vector as an arrow, the direction angle is the angle between the arrow and the positive x-axis, moving counterclockwise.

• Direction angles are typically given in degrees, like 30° in our example.
• They can be converted to radians, which is essential for calculations in trigonometry.

Knowing the direction angle allows us to apply trigonometric functions to find the vector's components. It helps us break down the vector along the x and y axes.

Magnitude of a Vector

The magnitude of a vector represents its length or size, essentially measuring how far it extends in space. In mathematical terms, it's always a non-negative number and is sometimes referred to as the vector's "norm."

• In our example, the magnitude of the vector \(\|\mathbf{v}\|\) is 5.
• This tells us the vector extends 5 units in the given direction from the origin.

Calculating a vector's components involves multiplying the magnitude by the cosine of the direction angle for the x-component, and the sine of the direction angle for the y-component. The magnitude stays constant, dictating how much the direction influences the overall displacement.

Conversion Between Degrees and Radians

Radians and degrees are two units for measuring angles. While degrees are more common in everyday life, radians are often used in mathematics, especially in trigonometry, because they simplify many equations.

• The conversion factor between degrees and radians is \(\pi/180\).
• In our exercise, we converted 30° to radians using this formula: \(\text{radians} = \frac{30}{180} \cdot \pi = \frac{\pi}{6}\).

Understanding this conversion is crucial, as many mathematical functions and equations, especially in calculus and physics, require angle measurements in radians. Remember, one full circle is \(2\pi\) radians, which is equivalent to 360 degrees.