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Let x be a vector in the plane containing vectors a=2i^-j^+k^ and b=i^+2j^-k^. If the vector x is perpendicular to (3i^+2j^-k^) and its projection on a is 1762, then the value of x2 is equal to _______.
MathematicsVector AlgebraJEE MainJEE Main 2021 (17 Mar Shift 2)
Solution:
2427 Upvotes Verified Answer
The correct answer is: 486

A plane containing two vectors can be expressed as a linear combination of the vectors.

Hence, let x=λa+μb (λ and μ are scalars).

x=i^(2λ+μ)+j^(2μ-λ)+k^(λ-μ)

Since x is perpendicular to 3i^+2j^-k^,

x·3i^+2j^-k^=0

i^(2λ+μ)+j^(2μ-λ)+k^(λ-μ)·3i^+2j^-k^=0

3(2λ+μ)+2(2μ-λ)-(λ-μ)=0

3λ+8μ=0   ...1

Also, the projection of x on a is 1762

x·a|a|=1762

i^(2λ+μ)+j^(2μ-λ)+k^(λ-μ)·2i^-j^+k^22+12+12=1762

2(2λ+μ)-(2μ-λ)+(λ-μ)6=1762

6λ-μ=51   ...2

On solving the equations (1) and (2), we get λ=8, μ=-3.

Thus, x=13i^-14j^+11k^

x=132+-142+112

|x|2=486.

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