Find the value of x such that the four-point with position vectors,"A"(3hat"i"+2hat"j"+hat"k"),"B" (4hat"i"+"x"hat"j"+5hat"k"),"c" (4hat"i"+2hat"j"-2hat"k")and"D"(6hat"i"+5hat"j"-hat"k")are coplaner. - Mathematics

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Sum

Find the value of x such that the four-point with position vectors,
`"A"(3hat"i"+2hat"j"+hat"k"),"B" (4hat"i"+"x"hat"j"+5hat"k"),"c" (4hat"i"+2hat"j"-2hat"k")`and`"D"(6hat"i"+5hat"j"-hat"k")`are coplaner.

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Solution

Let A, B, C, D be the given points. Then,

`vec("AB") = (4hat"i"+"x"hat"j"+5hat"k")-(3hat"i"+2hat"j"+hat"k")=hat"i"+("x"-2)hat"j"+4hat"k"`

`vec("AC") = (4hat"i"+2hat"j"-2hat"k")-(3hat"i"+2hat"j"+hat"k")=hat"i"+0hat"j"-3hat"k"`

`vec("AD") = (6hat"i"+2hat"j"-hat"k")-(3hat"i"+2hat"j"+hat"k")=3hat"i"+3hat"j"-2hat"k"`

The given points are coplanar if vectors `vec("AB") ,vec("AC") ,vec("AD")` are corplaner.
Therefore,
`[vec("AB")  vec("AC")  vec("AD")] = 0`

`⇒|(1,("x"-2),4),(1,0,-3),(3,3,-2)|=0`

⇒ 1 (0+9) - (x-2) (-2+9) + 4(3-0) = 0

⇒ 35 - 7x = 0

⇒ x =5

Hence, all the four points are coplanar for x =5.

Concept: Position Vector of a Point Dividing a Line Segment in a Given Ratio
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