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Three Dimensional Geometry

Short Answer Type

211. Find the equation of the plane through the points (0, – 1, 0), (2, 1, –1) and (1,1,1).

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212. Find the equation of the plane passing through the points (0. – 1, 0), (1, 1, 1) and (3, 3,0).

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Long Answer Type

213. Find the equation of the plane through the three points (1, 1, 1), (1, – 1, 1) and (–7, – 3,–5).

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Short Answer Type

214.

Find the vector equation of the plane passing through the point A(2, 2, –1), B(3, 4, 2) and C (7, 0, 6). Also find the cartesian equation of the plane.

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215. Find the vector equations of the plane passing through the points R(2, 5, –3), S(– 2, – 3, 5) and T(5, 3, – 3).

Here,    $straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 5 space straight j with hat on top space minus space 3 space straight k with hat on top comma space space straight b with rightwards arrow on top space equals space minus 2 space straight i with hat on top space minus space 3 space straight j with hat on top space minus space 5 space straight k with hat on top comma space space straight c with rightwards arrow on top space equals space 5 space straight i with hat on top space plus space 3 space straight j with hat on top space minus space 3 space straight k with hat on top$
where $straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top$ are position vectors  of R, S, T respectively.
The vector equation of plane is
$open parentheses straight r with rightwards arrow on top space minus space straight a with rightwards arrow on top close parentheses space. space open parentheses straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top close parentheses space cross times space left parenthesis straight c with rightwards arrow on top space minus space stack straight a right parenthesis with rightwards arrow on top space equals space 0$
or    $open square brackets straight r with rightwards arrow on top space minus space left parenthesis 2 space straight i with hat on top space plus space 5 space straight j with hat on top space minus space 3 space straight k with hat on top right parenthesis close square brackets. space space space open square brackets left parenthesis negative 4 space straight i with hat on top space minus space 8 space straight j with hat on top space minus space 2 space straight k with hat on top space right parenthesis cross times space left parenthesis 3 space straight i with hat on top space minus space space 2 space straight j with hat on top right parenthesis close square brackets space equals space 0$

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216. Find the equations of the planes that passes through three points:
(1, 1,–1), (6, 4, –5),(–4, –2, 3)

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217. Find the equations of the planes that passes through three points:
(1, 1, 0), (1, 2, 1), (–2, 2, –1)

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Long Answer Type

218. If from a point P (a, b, c) perpendiculars PA and PB are drawn to yz and zx-planes, then find the vector equation of the plane OAB.

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219. Find the equation of plane through the points (2, 2, 1), (9, 3, 6), and perpendicular to the plane 2x + 6y + 6z = 9.

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220. Find the equation of the plane through the points (2, 2, 1), (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z – 1 = 0.

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