Problem 44 - College Trigonometry

Question

For the given vector \(\vec{v}\), find the magnitude \(\|\vec{v}\|\) and an angle \(\theta\) with \(0 \leq \theta<360^{\circ}\) so that \(\vec{v}=\|\vec{v}\|\langle\cos (\theta), \sin (\theta)\rangle\) (See Definition 11.8.) Round approximations to two decimal places. $$ \vec{v}=\langle-4,3\rangle $$

Step-by-Step Solution

Answer

The magnitude is 5 and the angle is approximately 143.13°.

Step 1: Calculate the Magnitude of the Vector

To find the magnitude \( \| \vec{v} \| \) of the vector \( \vec{v} = \langle -4, 3 \rangle \), we use the formula: \[ \| \vec{v} \| = \sqrt{x^2 + y^2} \] Substitute \( x = -4 \) and \( y = 3 \) into the formula: \[ \| \vec{v} \| = \sqrt{(-4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \] Thus, the magnitude of \( \vec{v} \) is 5.

Step 2: Use Inverse Tangent to Find the Angle

The angle \( \theta \) can be found using the inverse tangent function: \[ \tan \theta = \frac{y}{x} = \frac{3}{-4} = -\frac{3}{4} \] Calculate \( \theta \) using \( \tan^{-1} \left( -\frac{3}{4} \right) \). Using a calculator in degrees mode gives: \[ \theta = \tan^{-1} \left( -\frac{3}{4} \right) \approx -36.87^{\circ} \]

Step 3: Adjust the Angle to the Correct Quadrant

Since the vector \( \langle -4, 3 \rangle \) lies in the second quadrant (where \( x < 0 \) and \( y > 0 \)), we need to adjust the angle \( \theta \) accordingly. Add \( 180^{\circ} \) to the angle: \[ \theta = -36.87^{\circ} + 180^{\circ} \approx 143.13^{\circ} \] This angle fits within the range \( 0 \leq \theta < 360^{\circ} \) for the second quadrant.

Key Concepts

Vector Magnitude

Understanding vector magnitude is essential in trigonometry. It tells you how long a vector is, similar to how the length of a line segment tells you the distance between its endpoints. Imagine \(\vec{v}\) as a combination of horizontal and vertical components. To calculate the vector's magnitude \(\| \vec{v} \|\), use the formula:

• \(\| \vec{v} \| = \sqrt{x^2 + y^2}\)

This formula is akin to the Pythagorean theorem, where \(x\) and \(y\) are the horizontal and vertical components of the vector, respectively.
For the vector \(\vec{v} = \langle -4, 3 \rangle\), substitute \(x = -4\) and \(y = 3\) into the magnitude formula:

• \| \vec{v} \| = \sqrt{(-4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \

This result tells us that the vector’s length is 5 units. Familiarity with vector magnitude gives you foundational insight into a vector's behavior in geometric space.

Angle Calculation

When it comes to determining the direction or angle of a vector in a coordinate plane, you need to calculate which angle \(\theta\) the vector makes with the positive x-axis. This angle is part of your vector's polar form, which represents vectors in terms of length and direction.
For a vector \(\vec{v} = \langle x, y \rangle\), the formula for calculating the tangent of the angle is:

• \(\tan \theta = \frac{y}{x}\)

To find \(\theta\) when the vector is \(\langle -4, 3 \rangle\), you take:

• \( \tan \theta = \frac{3}{-4} = -\frac{3}{4} \)

Use a calculator to get \(\theta\):

• \(\theta = \tan^{-1} \left( -\frac{3}{4} \right) \approx -36.87^{\circ}\)

However, note that \(\theta\) must fit within the standard range of angles, considering the quadrant in which the vector resides.

Inverse Tangent Function

The inverse tangent function, denoted as \(\tan^{-1}\) or \( ext{atan}\), is crucial when determining angles. It allows you to find an angle given the ratio of the opposite side to the adjacent side in a right triangle, which corresponds to the y and x components of a vector.
For vectors, the inverse tangent function helps translate the direction from component values to an actual angle. Think of it as converting a slope to an angle of inclination.

• Formula: \(\theta = \tan^{-1} \left( \frac{y}{x} \right)\)

After obtaining the acute angle, ensure it's adjusted based on the vector's quadrant. In our case:

• Since \(\vec{v} = \langle -4, 3 \rangle\) is in the second quadrant, add \(180^{\circ}\) to move from the negative angle to a positive one: \(\theta = -36.87^{\circ} + 180^{\circ} = 143.13^{\circ}\).

This adaptation respects the directional conventions of angle measures across coordinate quadrants, ensuring your final answer represents the true vector direction.