(i)
![\[A\left( 4,\text{ }-3,\text{ }-1 \right),\text{ }B\left( 5,\text{ }-7,\text{ }6 \right)\text{ }and\text{ }C\left( 3,\text{ }1,\text{ }-8 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-93f39660944efca93306ddaf999019cf_l3.png "Rendered by QuickLaTeX.com")
Given:
The points
Points
![\[A,\text{ }B\text{ }and\text{ }C\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-30bf1e81662bb73f4dfe4edf8ad36fd6_l3.png "Rendered by QuickLaTeX.com")
are collinear if
![\[AB\text{ }+\text{ }BC\text{ }=\text{ }AC\text{ }or\text{ }AB\text{ }+\text{ }AC\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-680c079fa1d49e34387e300e2ee647b2_l3.png "Rendered by QuickLaTeX.com")
![\[=\text{ }BC\text{ }or\text{ }AC\text{ }+\text{ }BC\text{ }=\text{ }AB\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-bb37f20c9d0e4d5c90904a272626e345_l3.png "Rendered by QuickLaTeX.com")
By using the formula,
The distance between any two points
![\[\left( a,\text{ }b,\text{ }c \right)\text{ }and\text{ }\left( m,\text{ }n,\text{ }o \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-35c34c22a15e72bb736c916fe956b30a_l3.png "Rendered by QuickLaTeX.com")
is given by,
![\[=~\surd 66\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0c22e901896968139f87add04942de55_l3.png "Rendered by QuickLaTeX.com")
∴The points
are collinear.
(ii)
![\[P\text{ }\left( 0,\text{ }7,\text{ }-7 \right),\text{}Q\text{}\left(1,\text{}4,\text{}-5 \right)\text{}and\text{}R\text{}\left( -1,\text{}10,\text{}-9\right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-baaaa27fbd76f05c45ec93d3c47a0e10_l3.png "Rendered by QuickLaTeX.com")
Given:
The points
![\[P\text{ }\left( 0,\text{ }7,\text{ }-7 \right),\text{ }Q\text{}\left(1,\text{}4,\text{}-5\right)\text{}and\text{}R\text{}\left( -1,\text{}10,\text{}-9 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b772a6b9717018cbabb8ff39d95582c2_l3.png "Rendered by QuickLaTeX.com")
Points
![\[P,\text{ }Q\text{ }and\text{ }R\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-196663053f18c63de9db91094cb6c6c1_l3.png "Rendered by QuickLaTeX.com")
are collinear if
![\[PQ\text{ }+\text{ }QR\text{ }=\text{ }PR\text{ }or\text{ }PQ\text{ }+\text{ }PR\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-14249fa2077dac500117e2bfc137e659_l3.png "Rendered by QuickLaTeX.com")
![\[=\text{ }QR\text{ }or\text{ }PR\text{ }+\text{ }QR\text{ }=\text{ }PQ\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-88f411b8153444055d07fa7c85677e7a_l3.png "Rendered by QuickLaTeX.com")
By using the formula,
Distance between any two points
is given by,
![\[=~\surd 14\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4098eb276464dbfbcd5d2571c5f469d0_l3.png "Rendered by QuickLaTeX.com")
∴The points
are collinear.