Problem 14 - APEX Calculus

Question

Find the measure of the angle between the two vectors in both radians and degrees. \(\vec{u}=\langle-2,1\rangle, \vec{v}=\langle 3,5\rangle\)

Step-by-Step Solution

Answer

The angle is approximately 1.6486 radians (94.478 degrees).

Step 1: Find the Dot Product

The dot product of two vectors \( \vec{u} = \langle -2, 1 \rangle \) and \( \vec{v} = \langle 3, 5 \rangle \) is given by: \[ \vec{u} \cdot \vec{v} = (-2) \cdot 3 + 1 \cdot 5 \]Calculating:\[ \vec{u} \cdot \vec{v} = -6 + 5 = -1 \]

Step 2: Compute Magnitudes of Each Vector

The magnitude of a vector \( \vec{u} = \langle a, b \rangle \) is given by:\[ \| \vec{u} \| = \sqrt{a^2 + b^2} \]For \( \vec{u} = \langle -2, 1 \rangle \):\[ \| \vec{u} \| = \sqrt{(-2)^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \]For \( \vec{v} = \langle 3, 5 \rangle \):\[ \| \vec{v} \| = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \]

Step 3: Calculate the Cosine of the Angle

The cosine of the angle \( \theta \) between two vectors is given by:\[ \cos(\theta) = \frac{\vec{u} \cdot \vec{v}}{\| \vec{u} \| \| \vec{v} \|} \]Substituting the values:\[ \cos(\theta) = \frac{-1}{\sqrt{5} \times \sqrt{34}} = \frac{-1}{\sqrt{170}} \]

Step 4: Calculate the Angle in Radians

The angle \( \theta \) in radians can be found using \( \theta = \cos^{-1} \left( \frac{-1}{\sqrt{170}} \right) \).Use a calculator to find:\( \theta \approx \cos^{-1}(-0.077) \approx 1.6486 \) radians.

Step 5: Convert the Angle to Degrees

To convert radians to degrees, use the formula: \[ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \]So,\[ \theta = 1.6486 \times \frac{180}{\pi} \approx 94.478 \text{ degrees} \]

Key Concepts

Dot Product

The dot product is a mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It is an essential concept in vector mathematics, especially when dealing with angles between vectors. To find the dot product of two vectors \( \vec{u} = \langle a_1, b_1 \rangle \) and \( \vec{v} = \langle a_2, b_2 \rangle \), you simply multiply their respective components and sum the results:

• \( \vec{u} \cdot \vec{v} = a_1 \cdot a_2 + b_1 \cdot b_2 \)

In our given problem, the vectors are \( \vec{u} = \langle -2, 1 \rangle \) and \( \vec{v} = \langle 3, 5 \rangle \). Therefore, the dot product is calculated as follows:

• \( (-2) \cdot 3 + 1 \cdot 5 = -6 + 5 = -1 \)

The dot product helps in determining whether two vectors are orthogonal (perpendicular) or to find the angle between them. If the dot product is zero, the vectors are orthogonal.

Magnitude of a Vector

The magnitude of a vector gives us the length or size of the vector, which is a way of representing the distance of the vector's tip from the origin in a coordinate system. This is computed using the Pythagorean theorem for each vector component.For a vector \( \vec{u} = \langle a, b \rangle \), its magnitude \( \| \vec{u} \| \) is calculated as:

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

Let's calculate the magnitudes of vectors provided in the problem:For \( \vec{u} = \langle -2, 1 \rangle \):

• \( \| \vec{u} \| = \sqrt{(-2)^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \)

For \( \vec{v} = \langle 3, 5 \rangle \):

• \( \| \vec{v} \| = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \)

Magnitude is crucial for understanding the scale of vectors and is used together with the dot product to understand relationships between vectors.

Cosine of an Angle

The cosine of the angle between two vectors is instrumental in finding the actual angle itself. This is because the cosine relates directly to the dot product and the magnitudes of the vectors. When you want to determine the angle \( \theta \) between two vectors, you use the formula:\[\cos(\theta) = \frac{\vec{u} \cdot \vec{v}}{\| \vec{u} \| \| \vec{v} \|}\]In our scenario, we have:

• Dot product: \( -1 \)
• Magnitude of \( \vec{u} \): \( \sqrt{5} \)
• Magnitude of \( \vec{v} \): \( \sqrt{34} \)

Substitute these values into the equation:

• \( \cos(\theta) = \frac{-1}{\sqrt{5} \times \sqrt{34}} = \frac{-1}{\sqrt{170}} \)

This fraction gives us the cosine of the angle in question. Using a calculator, you can find the cosine angle value, which we use in the next step to determine the angle in radians.

Radians to Degrees Conversion

Converting radians to degrees is a common task when working with angles. The reason for this conversion is often due to degrees being a more intuitive measure for angles compared to radians, especially in educational and applied settings. The conversion is done using the formula:\[\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\]In our problem, we found the angle \( \theta \) in radians to be approximately \( 1.6486 \) radians:

• \( \theta = 1.6486 \times \frac{180}{\pi} \approx 94.478 \text{ degrees} \)

This conversion allows you to express angles in a unit (degrees) that is commonly used and understood. Whether you are programming with angles, working on geometry problems, or reviewing maps, understanding how to switch between radians and degrees is an invaluable skill.