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Equation of a Plane Passing through Three Points

Find the vect...

Question

Find the vector and Cartesian equations of the plane passing through the points with position vectors
$3 \to i + 4 \to j + 2 \to k, 2 \to i - 2 \to j - \to k$ , and $7 \to i + \to j + \to k$

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Solution

Equation f the perpendicular to the plane

$\to n = ∣ ∣ ∣ ∣ ∣ \begin{matrix} \to i & \to j & \to k 1 & 6 & 3 - 4 & 3 & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$ $= - 3 \to i - 13 \to j + 27 \to k$

Equation of plane passing through three given points is given by:

$(\to n) \cdot (\to r - {\to r}_{o}) = 0$

$\Rightarrow (- 3 \to i - 13 \to j + 27 \to k) \cdot ((x - 3) \to i + (y - 4) \to j + (z - 2) \to k) = 0$

Cartesian form of plane :

$= - 3 (x - 3) - 13 (y - 4) + 27 (z - 2) = 0 = - 3 x - 13 y + 27 z + 7 = 0$

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Similar questions

Q. The vector equation of the plane through the point

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

⊥

to the line of intersection of the plane

\to r . (3 \to i - \to j + \to k) = 1

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

Q. Find the lengths of the following vectors :

3 \to i + 2 \to j - \to k

\sqrt{2} i - \sqrt{3} k

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

A

be vector parallel to line of intersection of planes

P_{1}

P_{2}

through origin.

P_{1}

is parallel to the vectors

2 \to j + 3 \to k

4 \to j - 3 \to k

P_{2}

\to j - \to k

3 \to i + 3 \to j

, then the angle between

A

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

Q. The unit vector which is orthogonal to the vector

\to a = 3 \to i + 2 \to j + 6 \to k

and is coplanar with the vectors

\to b = 2 \to i + \to j + \to k

\to c = \to i - \to j + \to k

P, Q, R

be points with position vectors

{\to r}_{1} = 3 \to i - 2 \to j - \to k, {\to r}_{2} = \to i + 3 \to j + 4 \to k

{\to r}_{3} = 2 \to i + \to j - 2 \to k

relative to an origin

O

. The distance of

P

O Q R

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