Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, -1) are collinear.
Last updated date: 23rd Mar 2023
•
Total views: 304.2k
•
Views today: 8.83k
Answer
304.2k+ views
Hint: Here to show that the 3 points are collinear we have to find the vectors of the given points and calculate its magnitude. If the points are collinear it means all points lie in a straight line.
Complete step-by-step answer:
As you know in question, we have to prove three points are collinear. First of all, you have to know the condition for collinearity.
Three points A(1, 2, 7), B(2, 6, 3) and C(3, 10, -1) are collinear
If and only if \[\left| {\overrightarrow {AB} } \right| + \left| {\overrightarrow {BC} } \right| = \left| {\overrightarrow {AC} } \right|\]
First find the vectors from \[\overrightarrow {AB} ,\overrightarrow {BC} ,\overrightarrow {AC} \]
$\overrightarrow {AB} = \left( {2 - 1} \right)\widehat i + \left( {6 - 2} \right)\widehat j + \left( {3 - 7} \right)\widehat k$
$\overrightarrow {AB} = \widehat i + 4\widehat j - 4\widehat k$
$\overrightarrow {BC} = \left( {3 - 2} \right)\widehat i + \left( {10 - 6} \right)\widehat j + \left( { - 1 - 3} \right)\widehat k$
$\overrightarrow {BC} = \widehat i + 4\widehat j - 4\widehat k$
$\overrightarrow {AC} = \left( {3 - 1} \right)\widehat i + \left( {10 - 2} \right)\widehat j + \left( { - 1 - 7} \right)\widehat k$
$\overrightarrow {AC} = 2\widehat i + 8\widehat j - 8\widehat k$
Now we have to calculate magnitude of these vectors \[\overrightarrow {AB} ,\overrightarrow {BC} , \overrightarrow {AC} \]
Magnitude of $\left| {\overrightarrow {AB} } \right| = \sqrt {{1^2} + {4^2} + {{\left( { - 4} \right)}^2}} $
$\left| {\overrightarrow {AB} } \right| = \sqrt {1 + 16 + 16} = \sqrt {33} $
Magnitude of $\left| {\overrightarrow {BC} } \right| = \sqrt {{1^2} + {4^2} + {{\left( { - 4} \right)}^2}} $
$\left| {\overrightarrow {BC} } \right| = \sqrt {1 + 16 + 16} = \sqrt {33} $
Magnitude of $\left| {\overrightarrow {AC} } \right| = \sqrt {{2^2} + {8^2} + {{\left( { - 8} \right)}^2}} $
$\left| {\overrightarrow {AC} } \right| = \sqrt {4 + 64 + 64} = \sqrt {132} = \sqrt {4 \times 33} $
$\left| {\overrightarrow {AC} } \right| = 2\sqrt {33} $
Now put the magnitude of these vectors In condition of collinearity.
$\left| {\overrightarrow {AB} } \right| + \left| {\overrightarrow {BC} } \right| = \sqrt {33} + \sqrt {33} = 2\sqrt {33} $
$\left| {\overrightarrow {AC} } \right| = 2\sqrt {33} $
Now you can easily see condition of collinearity satisfy
\[\left| {\overrightarrow {AB} } \right| + \left| {\overrightarrow {BC} } \right| = \left| {\overrightarrow {AC} } \right| = 2\sqrt {33} \]
Hence proved three point A(1, 2, 7), B(2, 6, 3) and C(3, 10, -1) are collinear
Note: Whenever you come to this type of problem, always apply the condition of collinearity. If some points are collinear it means all points lie in a straight line. It’s the geometrical application of collinearity.
Complete step-by-step answer:
As you know in question, we have to prove three points are collinear. First of all, you have to know the condition for collinearity.
Three points A(1, 2, 7), B(2, 6, 3) and C(3, 10, -1) are collinear
If and only if \[\left| {\overrightarrow {AB} } \right| + \left| {\overrightarrow {BC} } \right| = \left| {\overrightarrow {AC} } \right|\]
First find the vectors from \[\overrightarrow {AB} ,\overrightarrow {BC} ,\overrightarrow {AC} \]
$\overrightarrow {AB} = \left( {2 - 1} \right)\widehat i + \left( {6 - 2} \right)\widehat j + \left( {3 - 7} \right)\widehat k$
$\overrightarrow {AB} = \widehat i + 4\widehat j - 4\widehat k$
$\overrightarrow {BC} = \left( {3 - 2} \right)\widehat i + \left( {10 - 6} \right)\widehat j + \left( { - 1 - 3} \right)\widehat k$
$\overrightarrow {BC} = \widehat i + 4\widehat j - 4\widehat k$
$\overrightarrow {AC} = \left( {3 - 1} \right)\widehat i + \left( {10 - 2} \right)\widehat j + \left( { - 1 - 7} \right)\widehat k$
$\overrightarrow {AC} = 2\widehat i + 8\widehat j - 8\widehat k$
Now we have to calculate magnitude of these vectors \[\overrightarrow {AB} ,\overrightarrow {BC} , \overrightarrow {AC} \]
Magnitude of $\left| {\overrightarrow {AB} } \right| = \sqrt {{1^2} + {4^2} + {{\left( { - 4} \right)}^2}} $
$\left| {\overrightarrow {AB} } \right| = \sqrt {1 + 16 + 16} = \sqrt {33} $
Magnitude of $\left| {\overrightarrow {BC} } \right| = \sqrt {{1^2} + {4^2} + {{\left( { - 4} \right)}^2}} $
$\left| {\overrightarrow {BC} } \right| = \sqrt {1 + 16 + 16} = \sqrt {33} $
Magnitude of $\left| {\overrightarrow {AC} } \right| = \sqrt {{2^2} + {8^2} + {{\left( { - 8} \right)}^2}} $
$\left| {\overrightarrow {AC} } \right| = \sqrt {4 + 64 + 64} = \sqrt {132} = \sqrt {4 \times 33} $
$\left| {\overrightarrow {AC} } \right| = 2\sqrt {33} $
Now put the magnitude of these vectors In condition of collinearity.
$\left| {\overrightarrow {AB} } \right| + \left| {\overrightarrow {BC} } \right| = \sqrt {33} + \sqrt {33} = 2\sqrt {33} $
$\left| {\overrightarrow {AC} } \right| = 2\sqrt {33} $
Now you can easily see condition of collinearity satisfy
\[\left| {\overrightarrow {AB} } \right| + \left| {\overrightarrow {BC} } \right| = \left| {\overrightarrow {AC} } \right| = 2\sqrt {33} \]
Hence proved three point A(1, 2, 7), B(2, 6, 3) and C(3, 10, -1) are collinear
Note: Whenever you come to this type of problem, always apply the condition of collinearity. If some points are collinear it means all points lie in a straight line. It’s the geometrical application of collinearity.
Recently Updated Pages
Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE
