Assertion: Vecot `(hati+hatj+hatk)` is perpendicular to `(hati-2hatj+hatk)`
Reason: Two non-zero vectors are perpendicular if their dot product is equal to zero.

Options

A. If both Assertion & Reason are Tune & the Reason is a correct explanation of the Assertion. `
B. If both Assertion & Reason are True but Reason is not a correct explanation of the Assertion.
C. If Assertion is True but the Reason is False.
D. If both Assertion & Reason are false.

Answer

To determine whether the assertion and reason provided are correct, we will follow these steps:

### Step 1: Define the Vectors
Let:
- Vector **A** = \( \hat{i} + \hat{j} + \hat{k} \)
- Vector **B** = \( \hat{i} - 2\hat{j} + \hat{k} \)

### Step 2: Check the Dot Product
To check if the vectors are perpendicular, we need to calculate the dot product \( \mathbf{A} \cdot \mathbf{B} \).

The dot product formula for two vectors \( \mathbf{A} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \) and \( \mathbf{B} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \) is given by:
\[ \mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2 + a_3b_3 \]

For our vectors:
- \( a_1 = 1, a_2 = 1, a_3 = 1 \) (for vector A)
- \( b_1 = 1, b_2 = -2, b_3 = 1 \) (for vector B)

Now, substituting these values into the dot product formula:
\[ \mathbf{A} \cdot \mathbf{B} = (1)(1) + (1)(-2) + (1)(1) \]
\[ = 1 - 2 + 1 \]
\[ = 0 \]

### Step 3: Conclusion on Perpendicularity
Since the dot product \( \mathbf{A} \cdot \mathbf{B} = 0 \), it confirms that the vectors **A** and **B** are perpendicular.

### Step 4: Verify the Reason
The reason states that two non-zero vectors are perpendicular if their dot product is equal to zero. This is indeed a correct statement based on the properties of vectors.

### Final Conclusion
Both the assertion and the reason are true, and the reason correctly explains the assertion.

### Answer
Both assertion and reason are true, and the reason is the correct explanation of the assertion.

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