Using vector method, show that the points A(4, 5, 1), B(0, -1, -1), C(3, 9, 4) and
D(-4, 4, 4) are coplanar.
Given Points :
A ≡ (4, 5, 1)
B ≡ (0, -1, -1)
C ≡ (3, 9, 4)
D ≡ (-4, 4, 4)
To Prove : Points A, B, C & D are coplanar.
Formulae :
4) Position Vectors :
If A is a point with co-ordinates (a1, a2, a3)
then its position vector is given by,
5) Vectors :
If A & B are two points with position vectors ,
Where,
then vector is given by,
6) Scalar Triple Product:
If
Then,
7) Determinant :
Answer :
For given points,
A ≡ (4, 5, 1)
B ≡ (0, -1, -1)
C ≡ (3, 9, 4)
D ≡ (-4, 4, 4)
Position vectors of above points are,
Vectors are given by,
………eq(1)
………eq(2)
………eq(3)
Now, for vectors
= 4(15) – 6(21) + 2(33)
= 60 – 126 + 66
= 0
Hence, vectors are coplanar.
Therefore, points A, B, C & D are coplanar.
Note : Four points A, B, C & D are coplanar if and only if