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Question

A vector perpendicular to the plane containing the points A(1,-1,2), B(2,0,-1) and C(0,2,1) is


A

4i+8j-4k

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B

8i+4j+4k

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C

3i+j+2k

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D

i+j-k

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Solution

The correct option is B

8i+4j+4k


Explanation for the correct option:

Step 1: Find the Cross product of A and B.

Given: A(1,-1,2) and B(2,0,-1)

So, A×B=i^j^k^1-1220-1=-1·-1+2·0i^-1·-1-2·2j^+1·0--1·2k^=i^+5j^+2k^

Step 2: Find the Cross product of B and C.

Given: B(2,0,-1) and C(0,2,1)

So, B×C=i^j^k^20-1021=0·1-2·-1i^-2·1--1·0j^+2·2-0·0k^=2i^-2j^+4k^

Step 3: Find the Cross product of A and C.

Given: A(1,-1,2) and C(0,2,1)

So, C×A=i^j^k^0211-12=2·2-1·-1i^-0·2-1·1j^+0·-1-1·2k^=5i^+j^-2k^

Step 4: Find the dot product of the three vectors:

Use the cross products of the three vectors taken two at a time to find the dot product.

A·B·C=(i^+5j^+2k^)+(2i^-2j^+4k^)+(5i^+j^-2k^)=8i^+4j^+4k^ A·B·C=A×B+B×C+C×A

Hence, option (B) is correct.


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