Question

Prove that the four points 4i + 5j + k, -(j + k), 3i + 9j + 4k and -4i + 4j + 4k are coplanar.

Answer 1

Answer:

Yes, the given points are CO-PLANAR.

Step-by-step explanation:

The given points are A = 4i + 5j + k ,B =  -(j + k) = 0i - j - k , C = 3i + 9j + 4k and D = -4i + 4j + 4k

So

u = AB = ( 0 - 4 , - 1 - 5 , - 1 - 1 ) = ( - 4 , - 6 , - 2 )

v = AC = ( 3 - 4 , 9 - 5 , 4 - 1 ) = ( - 1 , 4 , 3 )

w = AD = ( -4 - 4 , 4 - 5 , 4 - 1 ) = ( -8 , - 1 , 3 )

The given points will be CO-PLANAR if

[ u v w ] = 0

So

       [ u v w ]

$=\left[\begin{array}{ccc}-4&-6&-2\\-1&4&3\\-8&-1&3\end{array}\right] \\\\=-4(12+3)-(-6)(-3+24)+(-2)(1+32)\\=-4(15)+6(21)-2(33)\\=-60+126-66\\=0$

Clearly  [ u v w ] = 0

So the given points are CO-PLANAR.

Answer 2

Answer:

Step-by-step explanation:

The given points are A = 4i + 5j + k ,B =  -(j + k) = 0i - j - k , C = 3i + 9j + 4k and D = -4i + 4j + 4k

So

u = AB = ( 0 - 4 , - 1 - 5 , - 1 - 1 ) = ( - 4 , - 6 , - 2 )

v = AC = ( 3 - 4 , 9 - 5 , 4 - 1 ) = ( - 1 , 4 , 3 )

w = AD = ( -4 - 4 , 4 - 5 , 4 - 1 ) = ( -8 , - 1 , 3 )

The given points will be CO-PLANAR if

[ u v w ] = 0

So

      [ u v w ]

Clearly  [ u v w ] = 0

So the given points are CO-PLANAR.