Problem 45 - 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. $$ \vec{v}=\langle-7,24\rangle $$

Step-by-Step Solution

Answer

Magnitude \( \|\vec{v}\| = 25 \) and angle \( \theta \approx 285.52^{\circ} \).

Step 1: Find the Magnitude of the Vector

To find the magnitude of the vector \(\vec{v} = \langle -7, 24 \rangle\), we use the formula \( \|\vec{v}\| = \sqrt{(-7)^2 + 24^2} \). Calculate this to find the magnitude.

Step 2: Compute the Magnitude

Calculate: \( \|\vec{v}\| = \sqrt{49 + 576} = \sqrt{625} = 25 \). Thus, the magnitude of the vector is 25.

Step 3: Determine the Angle Using components

Use the formula \( \theta = \arctan\left(\frac{y}{x}\right) \) where \(x = -7\) and \(y = 24\). So, \( \theta = \arctan\left(\frac{24}{-7}\right) \).

Step 4: Compute the Angle in Radians

Calculate \( \arctan\left(\frac{24}{-7}\right) \) using a calculator. This gives approximately \( \theta \approx -1.29249667 \) radians.

Step 5: Convert Radians to Degrees

Convert the radians to degrees by multiplying by \( \frac{180}{\pi} \). Thus, \( \theta \approx -1.29249667 \times \frac{180}{\pi} \approx -74.48^{\circ} \).

Step 6: Adjust the Angle to be Within 0 to 360 Degrees

Since \( \theta \) should be between \(0\) and \(360\) degrees, add \(360\) to \(-74.48^{\circ}\) to get \( \theta \approx 285.52^{\circ} \).

Step 7: Verify with the Definition

Check \(\vec{v} = \|\vec{v}\|\langle\cos(\theta), \sin(\theta)\rangle\). Calculate: \(\cos(285.52^{\circ}) \approx -0.28 \) and \(\sin(285.52^{\circ}) \approx 0.96\). Substitute back into \(\|\vec{v}\|\langle \cos(\theta), \sin(\theta) \rangle \) to confirm it matches \(\langle -7, 24 \rangle\) when rounded.

Key Concepts

Vector Magnitude

The magnitude of a vector, often denoted as \(\|\vec{v}\|\), represents its length or size. For a vector \(\vec{v} = \langle a, b \rangle\), the magnitude can be found using the Pythagorean theorem. Think of it as the distance from its initial point to its endpoint in a coordinate system.

Here’s the formula to find the magnitude:

• \( \|\vec{v}\| = \sqrt{a^2 + b^2} \)

Substitute the actual components of the vector into the formula to find the magnitude. For example, for \(\vec{v} = \langle -7, 24 \rangle\), it calculates like this:

• First, square the components: \((-7)^2 = 49\) and \(24^2 = 576\).
• Add these squares together: \(49 + 576 = 625\).
• Take the square root of this sum: \(\sqrt{625} = 25\).

Therefore, the magnitude of this vector is 25.

Angle Determination

Angle determination with respect to a vector involves finding the angle it makes with a standard position or another axis. This angle is crucial because it tells you the direction of the vector in a plane. You can easily find this angle using the components of the vector and trigonometric functions like tangent.

For a vector \(\vec{v} = \langle a, b \rangle\), the angle \(\theta\) with the \(x\)-axis can be found using the inverse tangent function (arctan), as follows:

• \( \theta = \arctan \left( \frac{b}{a} \right) \)

Keep in mind that tangent's range is limited, so the output might need adjustment depending on which quadrant your vector lies in.

For the example vector \(\vec{v} = \langle -7, 24 \rangle\), calculating \(\theta = \arctan \left( \frac{24}{-7} \right)\) results in an angle in radians. This might initially be negative or a small angle depending on the quadrant, and later needs converting to the suitable angle between \(0\) and \(360\) degrees.

Conversion Between Radians and Degrees

Sometimes your angle calculation gives you a value in radians, which is a way of measuring angles based on the radius of a circle. Although radians are mathematically elegant, degrees are more intuitive, especially when dealing with everyday problems or geometry in education.

To convert from radians to degrees, use this formula:

• \(\theta_{degrees} = \theta_{radians} \times \frac{180}{\pi}\)

This multiplication converts the angle value into a more familiar unit. For instance, if you have calculated that \(\theta = -1.29249667 \) radians, multiply it by \(\frac{180}{\pi}\) to find the angle in degrees, which would be roughly \(-74.48^\circ\).

Then, adjust this angle to ensure it falls within the standard range of \(0\) to \(360\) degrees. If the angle is negative, add \(360\) to bring it to the positive rotation direction of a full circle, obtaining approximately \(285.52^\circ\).

Component Form of Vectors

Vectors can be represented in component form, making them very versatile in computations and easy to visualize in a two-dimensional plane. A vector \(\vec{v}\) in component form is depicted by its horizontal component (\(x\)) and vertical component (\(y\)), such that \(\vec{v} = \langle x, y \rangle\).

The components of the vector help clarify its direction and magnitude, which aids in performing algebraic operations like addition, subtraction, and scaling on vectors. When understanding vectors, especially in physics or engineering, this component form is essential.

Due to the relationship with its magnitude and direction (angle), any vector can also be represented using trigonometry as \(\|\vec{v}\| \langle \cos(\theta), \sin(\theta) \rangle\), where \(\theta\) is the angle the vector makes with the positive \(x\)-axis. In this expression, \(\cos(\theta)\) gives proportion along the \(x\)-axis, and \(\sin(\theta)\) along the \(y\)-axis. Thus, it connects the abstract algebraic representation with geometrical visualization.