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Equation of a Plane Passing through Three Points

Equation of t...

Question

Equation of the plane through three points $A, B, C$ with position vectors $- 6 \to i + 3 \to j + 2 \to k, 3 \to i - 2 \to j + 4 \to k, 5 \to i + 7 \to j + 3 \to k$ is

A

\to r \cdot (\to i - \to j - 7 \to k) + 23 = 0

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B

\to r \cdot (\to i + \to j + 7 \to k) = 23

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C

\to r \cdot (\to i + \to j - 7 \to k) + 23 = 0

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D

\to r \cdot (\to i - \to j - 7 \to k) = 23

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Solution

The correct option is A $\to r \cdot (\to i - \to j - 7 \to k) + 23 = 0$
We have $\to A = - 6 \to i + 3 \to j + 2 \to k, \to B = 3 \to i - 2 \to j + 4 \to k, \to C = 5 \to i + 7 \to j + 3 \to k$
Thus $\to A B = 9 \to i - 5 \to j + 2 \to k, \to A C = 11 \to i + 4 \to j + \to k$
Normal vector perpendicular to plane $A B C$ is $\to n = \to A B \times \to A C$
$\Rightarrow \to n = ∣ ∣ ∣ ∣ ∣ \begin{matrix} \to i & \to j & \to k 9 & - 5 & 2 11 & 4 & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = - 13 \to i + 13 \to j + 91 \to k = 13 (- \to i + \to j + 7 \to k)$
Thus equation of plane $A B C$ is $(\to r - \to A) \cdot \to n = 0$
$\Rightarrow (x + 6) (- 1) + (y - 3) (1) + (z - 2) (7) = 0$
$\Rightarrow x + 6 - y + 3 - 7 z + 14 = 0$
$\Rightarrow (x - y - 7 z) + 23 = 0$
$\Rightarrow \to r \cdot (\to i - \to j - 7 \to k) + 23 = 0$
Hence, option 'A' is correct choice.

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Similar questions

Q. The ratio in which the plane

\to r \cdot (\to i - 2 \to j + 3 \to k) = 17

divides the line joining the points

- 2 \to i + 4 \to j + 7 \to k

3 \to i - 5 \to j + 8 \to k

Q. The distance between the line

\to r = 2 \to i - 2 \to j + 3 \to k + λ (\to i - \to j + 4 \to k)

\to r \cdot (\to i + 5 \to j + \to k) = 5

Q. The work done in moving a particle from the point

A

with position vector

2 \to i - 6 \to j + 7 \to k

B

with position vector

3 \to i - \to j - 5 \to k

\to F = \to i + 3 \to j - \to k

The shortest distance between the lines $\to r = 3 \to i + 5 \to j + 7 \to k + λ (\to i + 2 \to j + \to k) a n d \to r = - \to i - \to j - \to k + μ (7 \to i - 6 \to j + \to k) i s$

\to A = 2 \to i + \to k, \to B = \to i + \to j + \to k,

\to C = 4 \to i - 3 \to j + 7 \to k

Determine a vector

\to R

\to R \times \to B = \to C \times \to B

\to R . \to A = 0

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