Points whose position vectors are 2 -k- 3 -2 -k- and -4 -3 k- will

Let A= nbsp;2 î+ĵ k̂,B= 3 î 2 ĵ+k̂ and C=î+4 ĵ 3 k̂ AB=OB OA=3 î 2 ĵ+k̂ 2 î ĵ+k̂=î 3 ĵ+2 k̂ |AB|=√((1)^2+( 3)^2+(2)^2)=√(1+9+4)=√(14) units. BC=OC OB=î+4 ĵ 3 ...

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Standard IX

Mathematics

Equation of a Plane Passing through a Point and Perpendicular to a Given Vector

Points whose ...

Question

Points whose position vectors are $2^i +^j -^k, 3^i - 2^j +^k and^i + 4^j - 3^k$ will

Form an equilateral triangle

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Form a right triangle

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Form a scalene triangle

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Collinear

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Solution

The correct option is D Collinear
Let $\to A = 2^i +^j -^k, \to B = 3^i - 2^j +^k and \to C =^i + 4^j - 3^k$
$- - \to A B = - - \to O B - - - \to O A = 3^i - 2^j +^k - 2^i -^j +^k =^i - 3^j + 2^k$
$| - - \to A B | = \sqrt{(1)^{2} + (- 3)^{2} + (2)^{2}} = \sqrt{1 + 9 + 4} = \sqrt{14} units. B C = O C - O B =^i + 4^j - 3^k - 3^i + 2^j -^k = - 2^i + 6^j - 4^k$
$| - - \to B C | = \sqrt{(- 2)^{2} + (6)^{2} + (- 4)^{2}} = \sqrt{4 + 36 + 16} = \sqrt{56} = 2 \sqrt{14}$ units.
$- - \to A C = - - \to O C - - - \to O A =^i + 4^j - 3^k - 2^1 -^j +^k = -^1 + 3^j - 2^k$
$| - - \to A C | = \sqrt{(- 1)^{2} + (3)^{2} + (- 2)^{2}} = \sqrt{1 + 9 + 4} = \sqrt{14}$ units.
$∴ | - - \to A B | + | - - \to A C | = | - - \to B C |$
Hence $B, A$ and $C$ are collinear.

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Equation of a Plane Passing through a Point and Perpendicular to a Given Vector

Standard IX Mathematics

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Points whose position vectors are 2^i+^j−^k,3^i−2^j+^k and ^i+4^j−3^k will

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