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Show that the four points with position vectors 4iˆ + 8 ˆj +12kˆ, 2iˆ+ 4 ˆj + 6kˆ,3iˆ+ 5 ˆj + 4kˆ and 5iˆ+8 ˆj + 5kˆ are coplanar.

5   Show that the four points with position vectors 4\hat i + 8 \hat j +12\hat k, 2\hat i+ 4\hat j + 6\hat k,3\hat i+ 5\hat j + 4 \hat k and 5\hat i + 8 \hat j +5 \hat k  are coplanar.

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Given four position vectors,

\\A=4\hat i + 8 \hat j +12\hat k,\\ B=2\hat i+ 4\hat j + 6\hat k,\\C=3\hat i+ 5\hat j + 4 \hat k

And,

D=5\hat i + 8 \hat j +5 \hat k

now these will be coplanar when 

AB=(2-4)\hat i + (4-8)\hat j+(6-12)\hat k=-2\hat i-4\hat j-6\hat k

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

CD=(5-3)\hat i + (8-5)\hat j+(5-4)\hat k=2\hat i+3\hat j+\hat k

Now, let's calculate the vector triple product of these vectors

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

\begin{vmatrix} -2 &-4 &-6 \\ 1&1 & -2\\ 2&3 &1 \end{vmatrix}=-2(1+6)+4(1+4)-6(3-2)=0

Hence, the four points are coplanar.

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