दर्शाइए कि बिंदु `A(1, 2, 7), B(2, 6, 3)` और `C(3, 10, -1)` संरेख हैं |

Options

A. collinear
B. coplanar
C. non-collinear
D. None of these

Answer

A

Explanation

The given points are A(1, 2, 7), B(2, 6, 3) and C(3, 10, -1).
Now, `AB=PV" of "B - PV " of "A`
`vec(AB)=(2hati+6hatj+3hatk)-(1hati+2hatj+7hatk)`
`=hati+4hatj-4hatk`
`implies |AB|=sqrt((1)^(2)+(4)^(2)+(-4)^(2))`
`=sqrt(1+16+16)=sqrt(33)`
`BC=PV" of "C-PV" of "B`
`=(3hati+10hatj-1hatk)-(2hati+6hatj+3hatk)`
`=hati+4hatj-4hatk`
`implies |BC|=sqrt((1)^(2)+(4)^(2)+(-4)^(2))`
`=sqrt(1+16+16)=sqrt(33)`
`AC=PV" of "C-PV" of "A`
`vec(AC)=(3hati+10hatj-1hatk)-(1hati+2hatj+7hatk)`
`=2hati+8hatj-8hatk`
`implies |AC|=sqrt(2^(2)+8^(2)+(-8)^(2))=sqrt(4+64+64)`
`=sqrt(132)=2sqrt(33)=sqrt(33)+sqrt(33)`
`therefore |AC|=|AB|+|BC|`
Hence, the given points A, B and C are collinear.