Cosine Of Angle Between Vectors: Examples & Solutions

Hey guys! Let's break down how to find the cosine of the angle between two vectors. This is a super useful concept in linear algebra and physics, so let's get right to it. We'll tackle a couple of examples to make sure we've got it down.

(a) Vectors in 2D: $\bar{u} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $\bar{v} = \begin{pmatrix} 6 \\ 8 \end{pmatrix}$

Okay, so we have two vectors in two-dimensional space. Our mission is to find the cosine of the angle between them. The formula we're going to use is:

$\cos(\theta) = \frac{\bar{u} \cdot \bar{v}}{|\bar{u}| |\bar{v}|}$

Where:

• $\bar{u} \cdot \bar{v}$ is the dot product of the two vectors.
• $|\bar{u}|$ is the magnitude (or length) of vector $\bar{u}$.
• $|\bar{v}|$ is the magnitude of vector $\bar{v}$.

Let's break it down step by step:

1. Calculate the Dot Product $\bar{u} \cdot \bar{v}$

The dot product is calculated as follows:

$\bar{u} \cdot \bar{v} = (1)(6) + (2)(8) = 6 + 16 = 22$

So, the dot product of $\bar{u}$ and $\bar{v}$ is 22.

2. Calculate the Magnitude of $\bar{u}$

The magnitude of $\bar{u}$ is:

$|\bar{u}| = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5}$

3. Calculate the Magnitude of $\bar{v}$

The magnitude of $\bar{v}$ is:

$|\bar{v}| = \sqrt{(6)^2 + (8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10$

4. Plug Everything Into the Formula

Now we have all the pieces, so let's plug them into the formula:

$\cos(\theta) = \frac{22}{\sqrt{5} \cdot 10} = \frac{22}{10\sqrt{5}} = \frac{11}{5\sqrt{5}}$

To rationalize the denominator, we multiply the numerator and denominator by $\sqrt{5}$:

$\cos(\theta) = \frac{11\sqrt{5}}{5 \cdot 5} = \frac{11\sqrt{5}}{25}$

So, the cosine of the angle between the two vectors is $\frac{11\sqrt{5}}{25}$.

(b) Vectors in 3D: $\bar{u} = \begin{pmatrix} 1 \\ -3 \\ 7 \end{pmatrix}$ and $\bar{v} = \begin{pmatrix} 8 \\ -2 \\ -2 \end{pmatrix}$

Now let's tackle vectors in three-dimensional space. The process is essentially the same, but we have an extra component to deal with. Again, we'll use the formula:

$\cos(\theta) = \frac{\bar{u} \cdot \bar{v}}{|\bar{u}| |\bar{v}|}$

1. Calculate the Dot Product $\bar{u} \cdot \bar{v}$

For 3D vectors, the dot product is calculated as:

$\bar{u} \cdot \bar{v} = (1)(8) + (-3)(-2) + (7)(-2) = 8 + 6 - 14 = 0$

Wow, the dot product is 0! This tells us something special. We'll come back to that in a moment.

2. Calculate the Magnitude of $\bar{u}$

The magnitude of $\bar{u}$ is:

$|\bar{u}| = \sqrt{(1)^2 + (-3)^2 + (7)^2} = \sqrt{1 + 9 + 49} = \sqrt{59}$

3. Calculate the Magnitude of $\bar{v}$

The magnitude of $\bar{v}$ is:

$|\bar{v}| = \sqrt{(8)^2 + (-2)^2 + (-2)^2} = \sqrt{64 + 4 + 4} = \sqrt{72} = 6\sqrt{2}$

4. Plug Everything Into the Formula

Now we plug everything into the formula:

$\cos(\theta) = \frac{0}{\sqrt{59} \cdot 6\sqrt{2}} = 0$

Since the dot product is 0, the cosine of the angle between the vectors is 0. This means the angle between the vectors is 90 degrees. In other words, the vectors are orthogonal or perpendicular to each other. How cool is that?

Key Takeaways

• The cosine of the angle between two vectors can be found using the formula $\cos(\theta) = \frac{\bar{u} \cdot \bar{v}}{|\bar{u}| |\bar{v}|}$.
• The dot product is a scalar value calculated by multiplying corresponding components of the vectors and summing the results.
• The magnitude of a vector is its length, calculated using the square root of the sum of the squares of its components.
• If the dot product of two vectors is 0, the vectors are orthogonal (perpendicular).

Why is this important?

Understanding how to find the angle between vectors is crucial in various fields:

• Physics: Calculating work done by a force, analyzing projectile motion, and understanding electromagnetic fields.
• Computer Graphics: Determining lighting and shading effects, performing rotations and transformations.
• Machine Learning: Measuring the similarity between data points in high-dimensional spaces.
• Engineering: Analyzing structural forces and designing stable structures.

Tips for Success

• Practice, practice, practice! The more you work through examples, the more comfortable you'll become with the formulas and concepts.
• Draw diagrams. Visualizing the vectors can help you understand the geometry and relationships involved.
• Double-check your calculations. It's easy to make a small arithmetic error that can throw off your entire answer.
• Understand the underlying concepts. Don't just memorize the formulas. Make sure you understand why they work.

Conclusion

So there you have it! Finding the cosine of the angle between vectors is a straightforward process once you understand the formulas and concepts. Remember to calculate the dot product and magnitudes correctly, and you'll be golden. Keep practicing, and you'll master this skill in no time. Keep rocking!