`A=4i+4j-4k and B=3i+j +4k`, then angle between vectors A and B is

Options

A. `180^(@)`
B. `90^(@)`
C. `45^(@)`
D. `0^(@)`

Explanation

To find the angle between the vectors A and B, we can use the dot product formula. The dot product of two vectors A and B is given by:

\[ A \cdot B = |A| |B| \cos \theta \]

Where:
- \( A \cdot B \) is the dot product of vectors A and B,
- \( |A| \) and \( |B| \) are the magnitudes of vectors A and B respectively,
- \( \theta \) is the angle between the two vectors.

### Step 1: Calculate the dot product \( A \cdot B \)

Given:
\[ A = 4i + 4j - 4k \]
\[ B = 3i + j + 4k \]

The dot product \( A \cdot B \) can be calculated as follows:

\[ A \cdot B = (4)(3) + (4)(1) + (-4)(4) \]

Calculating each term:
- \( 4 \times 3 = 12 \)
- \( 4 \times 1 = 4 \)
- \( -4 \times 4 = -16 \)

Now, summing these results:
\[ A \cdot B = 12 + 4 - 16 = 0 \]

### Step 2: Use the dot product to find the angle

Since we have found that \( A \cdot B = 0 \), we can use the property of the dot product:

\[ A \cdot B = |A| |B| \cos \theta \]

If \( A \cdot B = 0 \), then:
\[ |A| |B| \cos \theta = 0 \]

This implies that \( \cos \theta = 0 \), which means:
\[ \theta = 90^\circ \]

### Conclusion

The angle between vectors A and B is \( 90^\circ \).