The angle between the two vectors `A=3 hati+4hatj+5hatk` and `B=3hati+4hatj-5hatk` is

Options

A. `60^@`
B. `45^@`
C. `90^@`
D. `30^@`

Explanation

To find the angle between the two vectors \( A = 3 \hat{i} + 4 \hat{j} + 5 \hat{k} \) and \( B = 3 \hat{i} + 4 \hat{j} - 5 \hat{k} \), we can use the dot product formula. Here are the steps to solve the problem:

### Step 1: Calculate the dot product of vectors A and B

The dot product \( A \cdot B \) is given by:

\[ A \cdot B = (3 \hat{i} + 4 \hat{j} + 5 \hat{k}) \cdot (3 \hat{i} + 4 \hat{j} - 5 \hat{k}) \]

Using the distributive property of the dot product:

\[ A \cdot B = 3 \cdot 3 + 4 \cdot 4 + 5 \cdot (-5) \]

Calculating each term:

\[ = 9 + 16 - 25 \]

\[ = 25 - 25 = 0 \]

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

The dot product can also be expressed in terms of the magnitudes of the vectors and the cosine of the angle \( \theta \) between them:

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

Since we found that \( A \cdot B = 0 \), we can set up the equation:

\[ 0 = |A| |B| \cos \theta \]

### Step 3: Analyze the equation

For the above equation to hold true, either \( |A| \) or \( |B| \) must be zero, or \( \cos \theta \) must be zero. Since both vectors \( A \) and \( B \) are non-zero vectors, we conclude:

\[ \cos \theta = 0 \]

### Step 4: Find the angle \( \theta \)

The cosine of an angle is zero at \( 90^\circ \). Therefore, we have:

\[ \theta = 90^\circ \]

### Conclusion

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

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