Problem 24 - Advanced Engineering Mathematics

Question

Find the angle \(\theta\) between the given vectors. $$ \mathbf{a}=\left\langle\frac{1}{2}, \frac{1}{2}, \frac{3}{2}\right\rangle, \mathbf{b}=\langle 2,-4,6\rangle $$

Step-by-Step Solution

Answer

The angle \( \theta \) is found using \( \theta = \cos^{-1}\left(\frac{8}{\sqrt{154}}\right) \).

Step 1: Recall the cosine formula for the angle between vectors

To find the angle \( \theta \) between two vectors \( \mathbf{a} \) and \( \mathbf{b} \), use the formula: \[ \cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{||\mathbf{a}|| \, ||\mathbf{b}||} \] where \( \mathbf{a} \cdot \mathbf{b} \) is the dot product of the vectors, and \( ||\mathbf{a}|| \) and \( ||\mathbf{b}|| \) are the magnitudes of the vectors.

Step 2: Calculate the dot product

The dot product \( \mathbf{a} \cdot \mathbf{b} \) is calculated as follows: \[ \mathbf{a} \cdot \mathbf{b} = \left(\frac{1}{2}\right)(2) + \left(\frac{1}{2}\right)(-4) + \left(\frac{3}{2}\right)(6) = 1 - 2 + 9 = 8 \] Thus, \( \mathbf{a} \cdot \mathbf{b} = 8 \).

Step 3: Calculate the magnitude of vector \( \mathbf{a} \)

The magnitude of vector \( \mathbf{a} \) is calculated as: \[ ||\mathbf{a}|| = \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^2 + \left(\frac{3}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{1}{4} + \frac{9}{4}} = \sqrt{\frac{11}{4}} = \frac{\sqrt{11}}{2} \]

Step 4: Calculate the magnitude of vector \( \mathbf{b} \)

The magnitude of vector \( \mathbf{b} \) is calculated as: \[ ||\mathbf{b}|| = \sqrt{2^2 + (-4)^2 + 6^2} = \sqrt{4 + 16 + 36} = \sqrt{56} = 2\sqrt{14} \]

Step 5: Substitute and solve for \( \cos(\theta) \)

Now, substitute the values into the cosine formula: \[ \cos(\theta) = \frac{8}{\left(\frac{\sqrt{11}}{2}\right)(2\sqrt{14})} = \frac{8}{\sqrt{154}} \]

Step 6: Calculate \( \theta \) using the inverse cosine

To find \( \theta \), use the inverse cosine function: \[ \theta = \cos^{-1}\left(\frac{8}{\sqrt{154}}\right) \] Use a calculator to get the angle in degrees or radians.

Key Concepts

Cosine Formula

To find the angle between two vectors, the cosine formula comes to our rescue. It's a very neat way to bridge the gap between vectors and angles. Imagine you have two vectors, and you want to know how far apart they are in terms of direction. That's where the cosine of the angle between them helps!

The cosine formula is expressed as:

• \( \cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{||\mathbf{a}|| \, ||\mathbf{b}||} \)

Here, \( \mathbf{a} \cdot \mathbf{b} \) is the dot product of the vectors, and \( ||\mathbf{a}|| \) and \( ||\mathbf{b}|| \) indicate the magnitudes of the vectors. By plugging the right values into this formula, you can find the cosine of the angle \( \theta \) between vectors \( \mathbf{a} \) and \( \mathbf{b} \). This is a foundational concept in understanding vector relationships.

Dot Product

The dot product is crucial for identifying how two vectors interact in space. It's a way to multiply vectors that results in a scalar (a single number) rather than another vector. Here's how you do it.

The dot product is calculated by multiplying the corresponding components of the vectors and adding them up:

• For vectors \( \mathbf{a} = \left\langle a_1, a_2, a_3 \right\rangle \) and \( \mathbf{b} = \left\langle b_1, b_2, b_3 \right\rangle \), the dot product \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \).

In our case, we calculate \[ \mathbf{a} \cdot \mathbf{b} = \left(\frac{1}{2}\right)(2) + \left(\frac{1}{2}\right)(-4) + \left(\frac{3}{2}\right)(6) = 8 \]This dot product tells us about the extent to which the vectors are "working together." The larger the value, the more aligned they are.

Vector Magnitude

The magnitude of a vector is like measuring its length, but in a multi-dimensional space. It tells us how long the vector is irrespective of its direction.

To find a vector's magnitude, we take the square root of the sum of the squares of its components:

• For vector \( \mathbf{a} = \left\langle a_1, a_2, a_3 \right\rangle \), the magnitude \( ||\mathbf{a}|| = \sqrt{a_1^2 + a_2^2 + a_3^2} \).

So for our vectors
- \( ||\mathbf{a}|| = \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^2 + \left(\frac{3}{2}\right)^2} = \frac{\sqrt{11}}{2} \)- \( ||\mathbf{b}|| = \sqrt{2^2 + (-4)^2 + 6^2} = 2\sqrt{14} \)

These values will help us later in calculating the cosine of the angle between the vectors.

Inverse Cosine

After you've determined the cosine of the angle between two vectors using the cosine formula, the next step is to find the actual angle. This is where the inverse cosine function, often symbolized as \( \cos^{-1} \), becomes useful.

Using the inverse cosine function allows you to convert the cosine value back to an angle measurement.
For our example, we use:

• \( \theta = \cos^{-1}\left(\frac{8}{\sqrt{154}}\right) \)

This gives you the angle \( \theta \) in radians or degrees, depending on your preference or requirement of the exercise. The inverse function essentially unlocks the angle from its cosine, giving you a clear geometric perspective. It's like unmasking the math to see the actual angle between your vectors.