We are given two vectors:

$\mathbf{a} = (0, 3, 4)$

$\mathbf{b} = (3, 0, 4)$

The cosine of the angle $\varphi$ between two vectors is given by the formula:

$\cos \varphi = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}$

Step 1: Compute the dot product $\mathbf{a} \cdot \mathbf{b}$

The dot product of two vectors $\mathbf{a} = (a_1, a_2, a_3)$ and $\mathbf{b} = (b_1, b_2, b_3)$ is calculated as:

$\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$

Substituting the given values:

$(0 \cdot 3) + (3 \cdot 0) + (4 \cdot 4) = 0 + 0 + 16 = 16$

Step 2: Compute the magnitudes of $\mathbf{a}$ and $\mathbf{b}$

The magnitude of a vector $\mathbf{a} = (a_1, a_2, a_3)$ is:

$|\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$

For $\mathbf{a} = (0, 3, 4)$:

$|\mathbf{a}| = \sqrt{0^2 + 3^2 + 4^2} = \sqrt{0 + 9 + 16} = \sqrt{25} = 5$

For $\mathbf{b} = (3, 0, 4)$:

$|\mathbf{b}| = \sqrt{3^2 + 0^2 + 4^2} = \sqrt{9 + 0 + 16} = \sqrt{25} = 5$

Step 3: Compute $\cos \varphi$

$\cos \varphi = \frac{16}{5 \times 5} = \frac{16}{25}$

Thus, the cosine of the angle between the vectors is:

$\cos \varphi = \frac{16}{25}$

Would you like a step-by-step explanation of how to find the angle $\varphi$ itself?

Here are five related questions:

1. How can we find the angle $\varphi$ from $\cos \varphi$?
2. What is the geometric interpretation of the dot product?
3. How do we determine if two vectors are perpendicular using the dot product?
4. What is the significance of the magnitude of a vector?
5. How can we generalize the cosine formula to n-dimensional vectors?

Tip: The dot product is useful for checking if vectors are orthogonal (perpendicular). If $\mathbf{a} \cdot \mathbf{b} = 0$, then the vectors are perpendicular!