Question

Show that four points with position vectors 4i^+8j^+12k^ , 2i^+4j^+6k^ , 3i^+5j^+4k^ and 5i^+8j^+5k^ are coplanar ?

Answer 1

For four vectors a,b,c,d to be coplaner ,
they should satisfy the condition of xa +yb+cz +td = 0
Here x,y,z,t are scalars, and they should satisfy x +y+z+ t =0
So let t = 1
And on solving using the above conditions:
☆》The middle answer see in my pic《《《

Answer 2

[AB AC AD] = 0, therefore the given vectors are coplanar.

• Let the given vectors be

               A = 4i + 8j + 12k

               B = 2i + 4j + 6k

               C = 3i + 5j + 4k

               D = 5i + 8j + 5k

                AB = (2i + 4j + 6k) - (4i + 8j + 12k)

                      = -2i - 4j - 6k

                AC = (3i + 5j + 4k) - (4i + 8j + 12k)

                      = -i - 3j - 8k

                AD = (5i + 8j + 5k) - (4i + 8j + 12k)

                      = i - 7k

               [ AB AC AD] =  -2(21-0)+4(7+8)-6(0+3)

                                    = - 42 + 60 - 18

                                    = 0

Refer attachment for first step in [AB AC AD]