`A=4i+4j-4k and B=3i+j4k`, 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 formula for 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 is calculated as follows:
\[ A \cdot B = (4i + 4j - 4k) \cdot (3i + j + 4k) \]

Calculating the dot product:
\[ A \cdot B = 4 \cdot 3 + 4 \cdot 1 + (-4) \cdot 4 \]
\[ = 12 + 4 - 16 \]
\[ = 0 \]

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

From the dot product formula:
\[ A \cdot B = |A| |B| \cos \theta \]
Since we found that \( A \cdot B = 0 \):
\[ 0 = |A| |B| \cos \theta \]

### Step 3: Determine the angle \( \theta \)

Since \( |A| \) and \( |B| \) are both non-zero (as they are not the zero vector), we can conclude:
\[ \cos \theta = 0 \]

This implies:
\[ \theta = 90^\circ \]

### Final Answer:
The angle between vectors A and B is \( 90^\circ \).