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To find the cosine of the angle \( \theta \) between vectors \( \mathbf{v} \) and \( \mathbf{w} \), we use the formula:

\[
\cos \theta = \frac{\mathbf{v} \cdot \mathbf{w}}{||\mathbf{v}|| \, ||\mathbf{w}||}
\]

### Step 1: Dot Product \( \mathbf{v} \cdot \mathbf{w} \)

The vectors are:

\[
\mathbf{v} = \begin{bmatrix} 5 \\ 5 \\ 3 \end{bmatrix}, \quad \mathbf{w} = \begin{bmatrix} 5 \\ 2 \\ 2 \end{bmatrix}
\]

The dot product \( \mathbf{v} \cdot \mathbf{w} \) is calculated as:

\[
\mathbf{v} \cdot \mathbf{w} = (5 \times 5) + (5 \times 2) + (3 \times 2) = 25 + 10 + 6 = 41
\]

### Step 2: Magnitude of \( \mathbf{v} \), \( ||\mathbf{v}|| \)

The magnitude of \( \mathbf{v} \) is:

\[
||\mathbf{v}|| = \sqrt{5^2 + 5^2 + 3^2} = \sqrt{25 + 25 + 9} = \sqrt{59}
\]

### Step 3: Magnitude of \( \mathbf{w} \), \( ||\mathbf{w}|| \)

The magnitude of \( \mathbf{w} \) is:

\[
||\mathbf{w}|| = \sqrt{5^2 + 2^2 + 2^2} = \sqrt{25 + 4 + 4} = \sqrt{33}
\]

### Step 4: Cosine of the angle

Now we compute \( \cos \theta \):

\[
\cos \theta = \frac{41}{\sqrt{59} \times \sqrt{33}} = \frac{41}{\sqrt{1947}}
\]

Approximating the value of \( \sqrt{1947} \):

\[
\sqrt{1947} \approx 44.13
\]

Thus,

\[
\cos \theta \approx \frac{41}{44.13} \approx 0.929
\]

So, \( \cos \theta \approx 0.929 \).

Would you like more details or have any questions?

### Related questions:
1. How do you calculate the angle if the cosine is given?
2. What is the geometric interpretation of the dot product in terms of angles?
3. How can you use the cross product to find the sine of the angle between two vectors?
4. What happens if two vectors are orthogonal (perpendicular)?
5. How does the cosine formula change in higher dimensions?

### Tip:
When calculating dot products and magnitudes, carefully check the components to avoid small calculation errors that can affect the final result.

Find the cosine of the angle between the vectors v = [5, 5, 3] and w = [5, 2, 2].

Learn how to find the cosine of the angle between the vectors v = [5, 5, 3] and w = [5, 2, 2] using the dot product formula. This problem involves key vector concepts, including calculating magnitudes and applying the Cauchy-Schwarz inequality. Suitable for students in Grades 11-12.

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