Question

50 points.... solve......

Answer 1

Let A = ( 1, - 1, 3 ), B = ( 2, - 4, 5 ) and C = ( 5, - 13, 11 ).

If the given points are in a Straight line, then

AB + BC = AC

AB = √( 2-1 ) ^2 + ( -4 + 1 )^2 + ( 5-3 )^2

AB = √ 1 + 9 + 4

AB = √ 14

AB = 2√5

BC = √ ( 5 - 2 ) ^2 + ( - 13 + 4 ) ^2 + ( 11 - 5 ) ^2

BC = √ 9 + 81 + 36 = 3 √ 14

AC = √ ( 5 - 1 )^2 + ( - 13 + 1 ) ^2 + ( 11 - 3 )^2

AC = √ 16 + 144 + 64

AC = 4 √ 14

AB + BC = AC

L. H. S = √ 14 + 3√ 14

L. H. S. = 4 √ 14 = AC

L. H. S. = R. H. S.

Hence, the given points are in a Straight line.

Answer 2

$\textbf{\huge{ANSWER}}$

Let ;-
A = ( 1 , -1 , 3)
B = ( 2 , -4 , 5)
C = ( 5 , -13, 11)

If , all these points lie on the straight lines than ;-

=> AC = AB + BC
=> AB = √( 2-1 )² + ( -4 + 1)² +( 5-3 )²
=> AB = √ 1 + 9 + 4
=> AB = √ 14
=> AB = 2√5
=> BC = √ ( 5 - 2 )² + ( - 13 + 4 )² + ( 11 - 5 )²
=> BC = √ 9 + 81 + 36 = 3 √ 14
=> AC = √ ( 5 - 1 )² + ( - 13 + 1 )² + ( 11 - 3 )²
=> AC = √ 16 + 144 + 64
=> AC = 4 √ 14
=> AB + BC = AC
=> L. H. S = √ 14 + 3√ 14
=> L. H. S. = 4 √ 14 = AC
=> L. H. S. = R. H. S.

Therefore,
The given points are in a Straight line.