If `a =2hati+3hatj+4hatk and b =4hati+3hatj +2hatk` ,find the angel between a and b.

Explanation

To find the angle between the vectors \( \mathbf{a} \) and \( \mathbf{b} \), we can use the formula for the cosine of the angle \( \theta \) between two vectors:

\[ \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|} \]

**Step 1: Calculate the dot product \( \mathbf{a} \cdot \mathbf{b} \)**

Given:
\[ \mathbf{a} = 2\hat{i} + 3\hat{j} + 4\hat{k} \]
\[ \mathbf{b} = 4\hat{i} + 3\hat{j} + 2\hat{k} \]

The dot product \( \mathbf{a} \cdot \mathbf{b} \) is calculated as follows:

\[ \mathbf{a} \cdot \mathbf{b} = (2)(4) + (3)(3) + (4)(2) \]
\[ = 8 + 9 + 8 = 25 \]

**Step 2: Calculate the magnitudes of \( \mathbf{a} \) and \( \mathbf{b} \)**

The magnitude of \( \mathbf{a} \) is given by:

\[ |\mathbf{a}| = \sqrt{(2^2) + (3^2) + (4^2)} = \sqrt{4 + 9 + 16} = \sqrt{29} \]

The magnitude of \( \mathbf{b} \) is given by:

\[ |\mathbf{b}| = \sqrt{(4^2) + (3^2) + (2^2)} = \sqrt{16 + 9 + 4} = \sqrt{29} \]

**Step 3: Substitute the values into the cosine formula**

Now substituting the values into the cosine formula:

\[ \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|} = \frac{25}{\sqrt{29} \cdot \sqrt{29}} = \frac{25}{29} \]

**Step 4: Calculate the angle \( \theta \)**

To find the angle \( \theta \), we take the inverse cosine:

\[ \theta = \cos^{-1}\left(\frac{25}{29}\right) \]

This gives us the angle between the vectors \( \mathbf{a} \) and \( \mathbf{b} \).

### Summary of Steps:
1. Calculate the dot product \( \mathbf{a} \cdot \mathbf{b} \).
2. Calculate the magnitudes \( |\mathbf{a}| \) and \( |\mathbf{b}| \).
3. Use the cosine formula to find \( \cos \theta \).
4. Calculate \( \theta \) using the inverse cosine.