In a tetrahedron, if two pairs of opposite edges are perpendicul

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 Multiple Choice QuestionsLong Answer Type

231.

Prove, using vectors, that the altitudes of a triangle are concurrent.
OR
Prove that the perpendicular from the vertices to the opposite sides of a triangle are concurrent. 

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232. Show that the diagonals of a rhombus bisect each other at right angles.
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233. In a triangle OAB , ∠AOB = 90°. If P and Q are the points of trisection of AB, prove that OP squared space plus space OQ squared space equals space 5 over 9 AB squared
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234. In a tetrahedron, if two pairs of opposite edges are perpendicular, then the third pair is also perpendicular to each other and the sum of the sequares of two opposite edges is the same for each pair.


Let ABCD be the tetrahedron. Take D as origin,
DA with rightwards arrow on top space equals space straight a with rightwards arrow on top comma space space space DB with rightwards arrow on top space equals space straight b with rightwards arrow on top comma space space DC with rightwards arrow on top space equals space straight c with rightwards arrow on top space so space that space 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 A, B, C  respectively with D as origin. Let DA and CB be perpendicular.

therefore space space DA with rightwards arrow on top. space CB with rightwards arrow on top space equals space 0 space space space or space space straight a with rightwards arrow on top. space left parenthesis straight b with rightwards arrow on top space minus space straight c with rightwards arrow on top right parenthesis space equals space 0 space space space space space... left parenthesis 1 right parenthesis
Let DB and AC be perpendicular.
        therefore space space DB with rightwards arrow on top. space AC with rightwards arrow on top space equals space 0 comma space space space or space space straight b with rightwards arrow on top. 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 space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis
Adding space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis comma space we space get space straight a with rightwards arrow on top. space left parenthesis straight b with rightwards arrow on top space minus space straight c with rightwards arrow on top right parenthesis space plus space straight b with rightwards arrow on top. space left parenthesis straight c with rightwards arrow on top space minus straight a with rightwards arrow on top right parenthesis space equals space 0
therefore space space space space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top. space straight c with rightwards arrow on top space plus space straight b with rightwards arrow on top space. space straight c with rightwards arrow on top space minus space straight b with rightwards arrow on top. space straight a with rightwards arrow on top space equals space 0 space space space or space space space straight c with rightwards arrow on top. space left parenthesis straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top right parenthesis space equals space 0
or space space DC with rightwards arrow on top. space AB with rightwards arrow on top space equals 0.

or space space DC with rightwards arrow on top space. space AB with rightwards arrow on top space equals space 0.

therefore space space space space space space space DC space is space perpendicular space to space AB.

Again,  DA squared plus CB squared space equals space straight a with rightwards arrow on top squared space plus space left parenthesis straight b with rightwards arrow on top space minus space straight c with rightwards arrow on top right parenthesis squared space equals space straight a with rightwards arrow on top squared plus straight b with rightwards arrow on top squared space plus space straight c with rightwards arrow on top squared space minus space 2 straight b with rightwards arrow on top. straight c with rightwards arrow on top
DB squared plus AC squared space equals straight b with rightwards arrow on top squared space plus space left parenthesis straight c with rightwards arrow on top space space minus straight a with rightwards arrow on top right parenthesis squared space equals space straight a with rightwards arrow on top squared space plus straight b with rightwards arrow on top squared space plus space straight c with rightwards arrow on top squared space minus space 2 space straight c with rightwards arrow on top space. straight a with rightwards arrow on top
DC squared plus AB squared space equals space straight c with rightwards arrow on top squared plus space left parenthesis straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top right parenthesis squared space equals straight a with rightwards arrow on top squared plus straight b with rightwards arrow on top squared space plus straight c with rightwards arrow on top squared minus space 2 straight a with rightwards arrow on top. space straight b with rightwards arrow on top
Now space from space left parenthesis 1 right parenthesis comma space space straight a with rightwards arrow on top. space straight b with rightwards arrow on top. space equals straight a with rightwards arrow on top. space straight c with rightwards arrow on top
From space left parenthesis 2 right parenthesis comma space space straight b with rightwards arrow on top. space straight c with rightwards arrow on top space equals space straight a with rightwards arrow on top. space straight b with rightwards arrow on top
therefore space space space we space have space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space minus straight b with rightwards arrow on top. space straight c with rightwards arrow on top space equals space straight c with rightwards arrow on top. space straight a with rightwards arrow on top
therefore space space DA squared plus CB squared space equals space DB squared plus AC squared space equals space DC squared plus AB squared
Hence space the space result.
 








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235.

Find straight lambda space and space straight mu if  left parenthesis 2 space straight i with hat on top space plus space 6 space straight j with hat on top space plus space 27 space straight k with hat on top right parenthesis space cross times left parenthesis straight i with hat on top space cross times space straight lambda space straight j with hat on top space plus space straight mu space straight k with hat on top right parenthesis space equals space 0 with rightwards arrow on top.

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 Multiple Choice QuestionsShort Answer Type

236.

Find open vertical bar straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top close vertical bar comma space space space if space space straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space plus straight j with hat on top space plus space 3 straight k with hat on top space and space straight b with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 5 space straight j with hat on top space minus space 2 space straight k with hat on top.

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237.

Find open vertical bar straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top close vertical bar  if  straight a with rightwards arrow on top space equals straight i with hat on top space minus space 7 space straight j with hat on top space plus space 7 space straight k with hat on top and straight b with rightwards arrow on top space equals space 3 straight i with hat on top space minus space 2 straight j with hat on top space plus space 2 straight k with hat on top

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238.

If straight a with rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space minus space 3 space straight k with hat on top space and space straight b with rightwards arrow on top space equals 3 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space 2 space straight k with hat on top comma space space find space open vertical bar 2 straight b with rightwards arrow on top cross times space straight a with rightwards arrow on top close vertical bar.

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 Multiple Choice QuestionsLong Answer Type

239.

If straight a with rightwards arrow on top space equals straight i with hat on top space minus straight j with hat on top space plus space 2 space straight k with hat on top comma space space straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space 3 space straight j with hat on top space minus space 4 space straight k with hat on top comma then show that straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space minus straight b with rightwards arrow on top space cross times space straight a with rightwards arrow on top

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 Multiple Choice QuestionsShort Answer Type

240. Find a unit vector perpendicular to both straight a with rightwards arrow on top space and space straight b with rightwards arrow on top if  straight a with rightwards arrow on top space equals space 4 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 2 space straight k with hat on top space space and space straight b 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

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