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Vectors vec(3i)-vec(2i)+veck and vec(2i)...

Vectors `vec(3i)-vec(2i)+veck` and `vec(2i)+vec(6j)+vec(mk)` will be perpendicular to each other if

A

m=8

B

m=6

C

m=3

D

m=1

Text Solution

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The correct Answer is:
To determine the value of \( m \) such that the vectors \( \vec{A} = 3\hat{i} - 2\hat{j} + \hat{k} \) and \( \vec{B} = 2\hat{i} + 6\hat{j} + m\hat{k} \) are perpendicular, we can follow these steps: ### Step 1: Understand the condition for perpendicularity Vectors \( \vec{A} \) and \( \vec{B} \) are perpendicular if their dot product is zero: \[ \vec{A} \cdot \vec{B} = 0 \] ### Step 2: Write the vectors The vectors are given as: \[ \vec{A} = 3\hat{i} - 2\hat{j} + \hat{k} \] \[ \vec{B} = 2\hat{i} + 6\hat{j} + m\hat{k} \] ### Step 3: Calculate the dot product The dot product \( \vec{A} \cdot \vec{B} \) is calculated as follows: \[ \vec{A} \cdot \vec{B} = (3)(2) + (-2)(6) + (1)(m) \] ### Step 4: Simplify the expression Calculating each term: \[ \vec{A} \cdot \vec{B} = 6 - 12 + m \] This simplifies to: \[ \vec{A} \cdot \vec{B} = -6 + m \] ### Step 5: Set the dot product to zero For the vectors to be perpendicular, we set the dot product equal to zero: \[ -6 + m = 0 \] ### Step 6: Solve for \( m \) Rearranging the equation gives: \[ m = 6 \] ### Final Answer Thus, the value of \( m \) for which the vectors are perpendicular is: \[ \boxed{6} \]
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