HOW TO PROVE FOUR VECTORS ARE COPLANAR

What is coplanar ?

The scalar triple product of three non-zero vectors is zero if, and only if, the three vectors are coplanar.

To check if the four vectors are coplanar, we have to follow the steps given below.

Step 1 :

Let the name of given vectors as A, B, C and D. If the given vectors are position vectors, then we give the name OA vector, OB vector, OC vector and OD vectors respectively.

Step 2 :

Combining these four vectors, create three vectors AB, AC and AD.

Step 3 :

Find the scalar triple product of these three vectors.

Step 4 :

If the scalar triple product is 0, then it must be a coplanar. Otherwise it is not.

Problem 1 :

Show that the four points (6, -7, 0), (16, -19, -4), (0, 3, -6), (2, -5,10) lie on a same plane.

Solution :

Let the points A = (6, -7, 0), B = (16, -19, -4), C = (0, 3, -6), D = (2, -5,10).

To show that the four points A, B, C and D lie on the plane, we have to prove that the three vectors

Finding AB vector :

Finding AC vector :

Finding AD vector :

Proving the given four vectors are coplanar :

So, the given four vectors are coplanar.

Problem 2 :

Show that four points with position vectors

are coplanar.

Solution :

Since the given vectors are position vectors,

Finding AB vector :

Finding AC vector :

Finding AD vector :

Checking if the given position vectors are coplanar :

= -4[12+3] + 6[-3+24] - 2[1+32]

= -4[15] + 6[21] - 2[33]

= -60 + 126 - 66

= -126 + 126

= 0

So, the given points are coplanar.

HOW TO PROVE FOUR VECTORS ARE COPLANAR

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