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  • Find x such that the four points A (3, 2, 1) B (4, x, 5), C (4, 2, –2) and D (6, 5, –1) are coplanar.

6   Find x such that the four points A (3, 2, 1) B (4, x, 5), C (4, 2, –2) and D (6, 5, –1) are coplanar.

Answers (1)

best_answer

Given

four points A = (3, 2, 1) , B= (4, x, 5), C= (4, 2, –2) and  D =(6, 5, –1) .

AB=(4-3)\hat i + (x-2)\hat j+(5-1)\hat k=\hat i+(x-2)\hat j+4\hat k

BC=(4-4)\hat i + (2-x)\hat j+(-2-5)\hat k=(2-x)\hat j-7\hat k

CD=(6-4)\hat i + (5-2)\hat j+(-1-(-2))\hat k=2\hat i+3\hat j+\hat k

These four points will be coplanar when the volume of the shape they make will be zero. which means 

\left [ AB,BC,CD \right ]=0

(\hat i+(x-2)\hat j+4\hat k)\cdot [((2-x)\hat j-7\hat k)\times(2\hat i+3\hat j+\hat k)]=0

(\hat i+(x-2)\hat j+4\hat k)\cdot [\begin{vmatrix} \hat i &\hat j &\hat k \\ 0&2-x & -7\\ 2&3 &1 \end{vmatrix}]=0

(\hat i+(x-2)\hat j+4\hat k)\cdot [\hat i (2-x+21)-\hat j(0-(-7))+\hat k(x-4)]=0

(23-x)-7(x-2)+4(x-4)=0

x=5

Hence the value of x is 5.

 

Posted by

Pankaj Sanodiya

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