The angle between the vectors `a = 2hati+2hatj-hatk and b = 6 hati - 3hatj +2hatk` is

Options

A. `cos^(-1)""3/11`
B. `cos^(-1)""2/11`
C. `cos^(-1)""4/21`
D. `cos^(-1)""3/22`

Answer

To find the angle between the vectors \( \mathbf{a} = 2\hat{i} + 2\hat{j} - \hat{k} \) and \( \mathbf{b} = 6\hat{i} - 3\hat{j} + 2\hat{k} \), we will use the formula for the dot product and the magnitudes of the vectors.

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

The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by:
\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]
where \( a_1, a_2, a_3 \) are the components of vector \( \mathbf{a} \) and \( b_1, b_2, b_3 \) are the components of vector \( \mathbf{b} \).

For our vectors:
- \( \mathbf{a} = (2, 2, -1) \)
- \( \mathbf{b} = (6, -3, 2) \)

Calculating the dot product:
\[ \mathbf{a} \cdot \mathbf{b} = (2)(6) + (2)(-3) + (-1)(2) = 12 - 6 - 2 = 4 \]

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

The magnitude of a vector \( \mathbf{v} = (v_1, v_2, v_3) \) is given by:
\[ |\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2} \]

Calculating the magnitude of \( \mathbf{a} \):
\[ |\mathbf{a}| = \sqrt{2^2 + 2^2 + (-1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]

Calculating the magnitude of \( \mathbf{b} \):
\[ |\mathbf{b}| = \sqrt{6^2 + (-3)^2 + 2^2} = \sqrt{36 + 9 + 4} = \sqrt{49} = 7 \]

### Step 3: Use the dot product to find \( \cos \theta \)

The relationship between the dot product and the angle \( \theta \) between two vectors is given by:
\[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta \]

Substituting the values we found:
\[ 4 = (3)(7) \cos \theta \]
\[ 4 = 21 \cos \theta \]
\[ \cos \theta = \frac{4}{21} \]

### Step 4: Calculate \( \theta \)

To find the angle \( \theta \), we take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{4}{21}\right) \]

### Final Answer

The angle \( \theta \) between the vectors \( \mathbf{a} \) and \( \mathbf{b} \) is:
\[ \theta = \cos^{-1}\left(\frac{4}{21}\right) \]

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