Problem 18 - Algebra and Trigonometry

Question

Find the angle between \(\mathrm{v}\) and \(\mathrm{w}\). Round to the nearest tenth of a degree. $$ \mathbf{v}=-2 \mathbf{i}+5 \mathbf{j}, \quad \mathbf{w}=3 \mathbf{i}+6 \mathbf{j} $$

Step-by-Step Solution

Answer

The angle between \(\mathbf{v}\) and \(\mathbf{w}\) is approximately 26.2°.

Step 1: Calculate the Dot Product

The dot product \(\mathbf{v} \cdot \mathbf{w}\) is calculated as (-2*3) + (5*6) = -6 + 30 = 24.

Step 2: Find the Magnitude of Each Vector

The magnitude of \(\mathbf{v}\) (denoted as \(||\mathbf{v}||\)) is \(\sqrt{(-2)^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29}\). The magnitude of \(\mathbf{w}\) (denoted as \(||\mathbf{w}||\)) is \(\sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45}\).

Step 3: Calculate the Cosine of the Angle

Divide the dot product by the product of the magnitudes to find the cosine of the angle \( \cos(\theta) = \frac{\mathbf{v} \cdot \mathbf{w}}{||\mathbf{v}|| \mid |\mathbf{w} ||} = \frac{24}{\sqrt{29} \times \sqrt{45}} \).

Step 4: Calculate the Angle

Finally, use the inverse cosine function to find the angle in radians, and convert to degrees by multiplying by \( \frac{180}{\pi} \). Round to the nearest tenth of a degree.

Key Concepts

Dot Product

The dot product is a way to multiply two vectors together, resulting in a scalar or a single number. It's a critical concept when finding the angle between two vectors. To calculate the dot product of vectors \( \mathbf{v} \) and \( \mathbf{w} \), we apply the formula:

• Multiply the corresponding components of the vectors together.
• Sum these products.

For vectors \( \mathbf{v} = -2 \mathbf{i} + 5 \mathbf{j} \) and \( \mathbf{w} = 3 \mathbf{i} + 6 \mathbf{j} \), the dot product is \((-2 \times 3) + (5 \times 6) = -6 + 30 = 24\).
So, the dot product of these vectors is 24. This value represents how much one vector extends in the direction of another vector.

Vector Magnitude

The magnitude of a vector, also known as its length, is a measure of how long a vector is in space. It's a fundamental concept used when computing the angle between vectors.
To find the magnitude, use the formula:

• Take the square root of the sum of the squares of its components.

For vector \( \mathbf{v} = -2 \mathbf{i} + 5 \mathbf{j} \), the magnitude \( ||\mathbf{v}|| \) is \( \sqrt{(-2)^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29} \).
Similarly, for vector \( \mathbf{w} = 3 \mathbf{i} + 6 \mathbf{j} \), the magnitude \( ||\mathbf{w}|| \) equals \( \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} \). These magnitudes are crucial when calculating the cosine of the angle between these vectors.

Inverse Cosine Function

The inverse cosine function is essential in determining angles when you know the cosine value. We often use it after calculating the cosine of the angle in vector problems.
Here's how it works:

• First, find the cosine of the angle using the identity \( \cos(\theta) = \frac{\mathbf{v} \cdot \mathbf{w}}{||\mathbf{v}|| \, ||\mathbf{w}||} \).
• For our vectors, this is \( \frac{24}{\sqrt{29} \times \sqrt{45}} \).

After finding this cosine value, use the inverse cosine function (often denoted as \( \cos^{-1} \)) to find the angle \( \theta \) in radians.
This function effectively 'undoes' the cosine function, providing the angle whose cosine is the computed value.

Radians to Degrees Conversion

When calculating angles in mathematics, we often switch between radians and degrees. Degrees are more intuitive, while radians are used frequently in calculus and trigonometry.
You convert radians to degrees using the conversion factor \( \frac{180}{\pi} \):

• Multiply the angle in radians by \( \frac{180}{\pi} \).

This conversion is crucial when presenting results in a format more commonly used in practical scenarios.
So, after you find the angle in radians using the inverse cosine function, multiply the result by \( \frac{180}{\pi} \) to get the angle in degrees.
Finally, round this degree value to the nearest tenth for precision in your calculations.