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Find the vector equation of the plane passing through points 4

Question

Find the vector equation of the plane passing through points $4 i - 3 j - k, 3 i + 7 j - 10 k$ and $2 i + 5 j - 7 k$ and show that the point $i + 2 j - 3 k$ lies in the plane.

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Sebastian Discussed

Find the vector equation of the plane passing through points

4 i - 3 j - k, 3 i + 7 j - 10 k

and

2 i + 5 j - 7 k

and show that the point

i + 2 j - 3 k

lies in the plane.

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Text solutionVerified

Points are (4,-3,-1);(3,7,-10);(2,5,-7)
Equation of plane will be

∣ ∣ x - 4 x - 3 x - 2 y + 3 y - 7 y - 5 z + 1 z + 10 z + 7 ∣ ∣ = 0

R_{2} \to R_{2} - R_{1} R_{3} \to R_{3} - R_{1}

∣ ∣ x - 4 12 y + 3 - 10 - 8 z + 1 96 ∣ ∣ = 0

(x - 4) ∣ ∣ - 10 - 8 96 ∣ ∣ - (y + 3) ∣ ∣ 1296 ∣ ∣ + (z + 1) ∣ ∣ 12 - 10 - 8 ∣ ∣ = 0

(x - 4) 12 - (y + 3) (- 12) + (z + 1) (12) - 0 x - 4 + y + 3 + z + 1 = 0 x + y + z = 0

For points

(1,, 2, - 3)

that is

i + 2 j - 3 k

1 + 2 - 3 - 0

0 = 0

LHS=RHS
Therefore,

i + 2 j - 3 k

lies on plane

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Question Text	Find the vector equation of the plane passing through points $4 i - 3 j - k, 3 i + 7 j - 10 k$ and $2 i + 5 j - 7 k$ and show that the point $i + 2 j - 3 k$ lies in the plane.
Answer Type	Text solution:1
Upvotes	150