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Home » Using distance formula prove that the following points are collinear: (i) A(4, -3, -1), B(5, -7, 6) and C(3, 1, -8) (ii) P(0, 7, -7), Q(1, 4, -5) and R(-1, 10, -9)
Using distance formula prove that the following points are collinear: (i) A(4, -3, -1), B(5, -7, 6) and C(3, 1, -8) (ii) P(0, 7, -7), Q(1, 4, -5) and R(-1, 10, -9)
CBSE | Class 11 | Exercise 28.2 | Introduction to Three Dimensional Coordinate Geometry | Maths | RD Sharma
Using distance formula prove that the following points are collinear: (i) A(4, -3, -1), B(5, -7, 6) and C(3, 1, -8) (ii) P(0, 7, -7), Q(1, 4, -5) and R(-1, 10, -9)

(i) 

    \[A\left( 4,\text{ }-3,\text{ }-1 \right),\text{ }B\left( 5,\text{ }-7,\text{ }6 \right)\text{ }and\text{ }C\left( 3,\text{ }1,\text{ }-8 \right)\]

Given:

The points

    \[A\left( 4,\text{ }-3,\text{ }-1 \right),\text{ }B\left( 5,\text{ }-7,\text{ }6 \right)\text{ }and\text{ }C\left( 3,\text{ }1,\text{ }-8 \right)\]

Points

    \[A,\text{ }B\text{ }and\text{ }C\]

are collinear if

    \[AB\text{ }+\text{ }BC\text{ }=\text{ }AC\text{ }or\text{ }AB\text{ }+\text{ }AC\]

    \[=\text{ }BC\text{ }or\text{ }AC\text{ }+\text{ }BC\text{ }=\text{ }AB\]

By using the formula,

The distance between any two points

    \[\left( a,\text{ }b,\text{ }c \right)\text{ }and\text{ }\left( m,\text{ }n,\text{ }o \right)\]

is given by,

RD Sharma Solutions for Class 11 Maths Chapter 28 – image 8

    \[=~\surd 66\]

RD Sharma Solutions for Class 11 Maths Chapter 28 – image 9

∴The points

    \[A,\text{ }B\text{ }and\text{ }C\]

are collinear.

(ii)

    \[P\text{ }\left( 0,\text{ }7,\text{ }-7 \right),\text{}Q\text{}\left(1,\text{}4,\text{}-5 \right)\text{}and\text{}R\text{}\left( -1,\text{}10,\text{}-9\right)\]

Given:

The points

    \[P\text{ }\left( 0,\text{ }7,\text{ }-7 \right),\text{ }Q\text{}\left(1,\text{}4,\text{}-5\right)\text{}and\text{}R\text{}\left( -1,\text{}10,\text{}-9 \right)\]

Points

    \[P,\text{ }Q\text{ }and\text{ }R\]

are collinear if

    \[PQ\text{ }+\text{ }QR\text{ }=\text{ }PR\text{ }or\text{ }PQ\text{ }+\text{ }PR\]

    \[=\text{ }QR\text{ }or\text{ }PR\text{ }+\text{ }QR\text{ }=\text{ }PQ\]

By using the formula,

Distance between any two points

    \[\left( a,\text{ }b,\text{ }c \right)\text{ }and\text{ }\left( m,\text{ }n,\text{ }o \right)\]

is given by,

RD Sharma Solutions for Class 11 Maths Chapter 28 – image 10

    \[=~\surd 14\]

RD Sharma Solutions for Class 11 Maths Chapter 28 – image 11

∴The points

    \[P,\text{ }Q\text{ }and\text{ }R\]

are collinear.

i

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