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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.

Here, $straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 2 space straight j with hat on top space minus space straight k with hat on top comma space space space space straight b with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 2 space straight k with hat on top comma space space straight c with rightwards arrow on top space equals space 7 space straight i with hat on top space plus space 6 space straight k with hat on top$
The vector equations of plane is
$open parentheses straight r with rightwards arrow on top space minus space straight a with rightwards arrow on top close parentheses. space space space left parenthesis straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top right parenthesis space cross times space left parenthesis straight c with rightwards arrow on top space minus space straight a with rightwards arrow on top right parenthesis space equals space 0$
or $open square brackets straight r with rightwards arrow on top space minus space left parenthesis 2 space straight i with hat on top space plus space 2 space straight j with hat on top space minus space straight k with hat on top right parenthesis close square brackets. space space open square brackets left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis space cross times left parenthesis 5 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space 7 space straight k with hat on top right parenthesis close square brackets space equals space 0$
The Cartesian equation of plane is
$open vertical bar table row cell straight x minus 2 end cell cell straight y minus 2 end cell cell straight z plus 1 end cell row 1 2 3 row 5 cell negative 2 end cell 7 end table close vertical bar space equals space 0$

or $left parenthesis straight x minus 2 right parenthesis space open vertical bar table row 2 3 row cell negative 2 end cell 7 end table close vertical bar space minus space left parenthesis straight y minus 2 right parenthesis space open vertical bar table row 1 3 row 5 7 end table close vertical bar space plus space left parenthesis straight z plus 1 right parenthesis space open vertical bar table row 1 2 row 5 cell negative 2 end cell end table close vertical bar space equals space 0$

or (or – 2) (14 + 6) – (y – 2) (7 – 15) + (z + 1) (–2 – 10) = 0
or 20 (x – 2) + 8 (y – 2) – 12 (z + 1) = 0
or 5 (x – 2) + 2 (y – 2) – 3 (z + 1) = 0
or 5 x – 10 + 2y – 4 – 3 z – 3 = 0
or 5 x + 2 y – 3 z – 17 = 0

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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).

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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.

100 Views

Previous Year Papers

Test Series

Test Yourself

Short Answer Type

Long Answer Type

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.

Long 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.