The angles between the two vectors `vecA=3hati+4hatj+5hatk` and `vecB=3hati+4hatj-5hatk` will be

Options

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

Explanation

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

### Step 1: Write down the vectors
We have:
\[ \vec{A} = 3\hat{i} + 4\hat{j} + 5\hat{k} \]
\[ \vec{B} = 3\hat{i} + 4\hat{j} - 5\hat{k} \]

### Step 2: Calculate the dot product \(\vec{A} \cdot \vec{B}\)
The dot product of two vectors \(\vec{A}\) and \(\vec{B}\) is given by:
\[ \vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z \]
Substituting the components:
\[ \vec{A} \cdot \vec{B} = (3)(3) + (4)(4) + (5)(-5) \]
Calculating each term:
\[ = 9 + 16 - 25 \]
\[ = 25 - 25 = 0 \]

### Step 3: Use the dot product to find the angle
The dot product is also related to the angle \(\theta\) between the vectors:
\[ \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta \]
Since we found that \(\vec{A} \cdot \vec{B} = 0\), we can write:
\[ 0 = |\vec{A}| |\vec{B}| \cos \theta \]
This implies:
\[ \cos \theta = 0 \]

### Step 4: Solve for \(\theta\)
The angle \(\theta\) for which \(\cos \theta = 0\) is:
\[ \theta = 90^\circ \]

### Conclusion
The angle between the two vectors \(\vec{A}\) and \(\vec{B}\) is \(90^\circ\).

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