Which of the following are equations for the plane passing through the points $P (1, 1, - 1), Q (3, 0, 2)$ and $R (- 2, 1, 0) ?$

(2^i - 3^j + 3^k) \cdot ((x + 2)^i + (y - 1)^j + z^k) = 0

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x = 3 - t, y = - 11 t, z = 2 - 3 t

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(x + 2) + 11 (y - 1) = 3 x

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(2^i -^j + 3^k) \times (- 3^i +^k) \cdot ((x + 2)^i + (y - 1)^j + z^k) = 0

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Solution

The correct option is B $(2^i -^j + 3^k) \times (- 3^i +^k) \cdot ((x + 2)^i + (y - 1)^j + z^k) = 0$
Given points are $P (1, 1, - 1), Q (3, 0, 2)$ and $R (- 2, 1, 0)$
Normal vector perpendicular to $P Q R$ is given by $(\to Q - \to P) \times \to Q = (2^i -^j + 3^k) \times (3^i + 2^k)$
Therefore equation of plane $P Q R$ is,
$(2^i -^j + 3^k) \times (3^i + 2^k) \cdot ((x + 2)^i + (y - 1)^j + z^k) = 0$

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Similar questions

If $\to r \times \to b = \to c \times \to b$ and $\to r . \to a = 0$ where $\to a = 2^i + 3^j -^k, \to b = 3^i -^j +^k$ and $\to c =^i +^j + 3^k$ , then $\to r$ is equal to

Q. Given the vertices of triangle by position vectors

^i +^j +^k,^i +^k a n d^j +^k

the centroid and Incentre of the triangle will be given by

Q. if

^r = 3^i + 1^j - 5 k,^a = 2^i -^j +^k,^b =^i + 3^j - 2^k

and

^c = - 2^i +^j - 3^k

such that

^r = l^a + m^b + n^n

then

\begin{matrix} Column I & Column II a & The distance between the line \to r = (2^i - 2^j + 3^k) + λ (^i -^j + 4^k) & p & \frac{25}{3 \sqrt{14}} and plane \to r . (^i + 5^j +^k) = 5 b & The distance between parallel planes \to r . (2^i -^j + 3^k) = 4 and & q & \frac{13}{7} \to r . (6^i - 3^j + 9^k) + 13 = 0 is c & The distance of a point (2, 5, -3) from the plane \to r . (6^i - 3^j + 2^k) = 4 & r & \frac{10}{3 \sqrt{3}} is d & The distance of the point (1, 0, -3) from the plane x - y - z - 9 = 0 & s & 7 measured parallel to line \frac{x - 2}{2} = \frac{y + 2}{3} = \frac{z - 6}{- 6} \end{matrix}

Then which of the following is correct ?

Q. The equation of the plane passing through the line of intersection of the planes

\to r \cdot (2^i - 3^j + 4^k) = 1

and

\to r \cdot (^i -^j) + 4 = 0

and perpendicular to the plane

\to r \cdot (2^i -^j -^k) + 4 = 0,

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Which of the following are equations for the plane passing through the points P(1,1,−1),Q(3,0,2) and R(−2,1,0)?

Which of the following are equations for the plane passing through the points $P (1, 1, - 1), Q (3, 0, 2)$ and $R (- 2, 1, 0) ?$