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I only know how to show that $2$ vectors are collinear, but for $3 $ vectors I only know how to prove coplanarity.

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3 Answers

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A similar problem is the determining if three points are collinear within a plane.

Given points a, b and c form the line segments ab, bc and ac. If ab + bc = ac then the three points are collinear.

The line segments can be translated to vectors ab, bc and ac where the magnitude of the vectors are equal to the length of the respective line segments mentioned.

By example of the points you've given in response to Naveen.

a(2, 4, 6) b(4, 8, 12) c(8, 16, 24)

$$\overline{ab} = \sqrt[]{56}$$ $$\overline{bc} = \sqrt[]{224}$$ $$\overline{ac} = \sqrt[]{504}$$

$$\overline{ab} + \overline{bc} = \overline{ac}$$

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For any vectors x,y,z $||x-y||\leq ||x-z||+||y-z||$ and equality hold iff all three are collinear.

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Ok see here..... Since 'use vectors to' is pretty vauge, you can do this a bunch of ways.

For example, you could use the dot product ((b⃗ −c⃗ )⋅(a⃗ −b⃗ )) / (|b⃗ −c⃗ | * |a⃗ −b⃗ |) = ± 1

or the cross product (b⃗ − c⃗ ) × (a⃗ −b⃗ )=0⃗

Actually you may proceed in a general co-ordinate geometry process also...... let us consider that the 3 vectors if we poit them into co-ordinate system we will get 3 points.....

let us consider they are: (a,b) , (c,d) , (e,f).. So, you will have 3 points on the co-ordinates......ok.

now you, find the equations of the straight line joining the points (a,b) and (c,d) and do the same for the points (c,d) , (e,f) ----- if these 3 points are co-linear then you will get the same equation of straight line!!

OR, You can proceed in this way: first measure the SLOPE of the line joining the points (a,b) and (e,f) that is : (f - b) / (e - a) .

and then measure the SLOPE of the line joining the points (a,b) and (c,d) that is : (d - b) / (c - a) .

 If the pints A(a,b) , B(c,d) , C(e,f) are co-linear then the measured slope will be equal

OR, You can find the equation of the st. line joining the points A and B and then you can find the equation of the st. line joining the points B and C then you find the angle between AB st.line and BC st.line...by applying the co-ordinate geometry formula (that you can easily get from your text book) and see whether the angle between the straight lines is 180deg. or not....if it is 180deg. you can easily say that the points A , B , C are co-linear.......

Hope the discussion will help you.best of luck.

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