If vector a, vector b, vector c are non- coplanar vectors, prove that the points having the following position vectors are collinear:

Question

If \(\vec a,\vec b,\vec c\) are non- coplanar vectors, prove that the points having the following position vectors are collinear:  \(\vec a+\vec b+\vec c,\) \(4\vec a+3\vec b,\) \(10\vec a+7\vec b-2\vec c\)

Answer 1

Let us understand that, two more points are said to be collinear if they all lie on a single straight line.

Given that, \(\vec a,\vec b\) and \(\vec c\) are non-coplanar vectors.

 And we know that, vectors that do not lie on the same plane or line are called non-coplanar vectors.

To Prove: \(\vec a+\vec b+\vec c,\) \(4\vec a+3\vec b\) and \(10\vec a+7\vec b-2\vec c\) are collinear.

Proof: Let the points be A, B and C.

Then,

So, in this case if we prove that \(\vec{AB}\) and \(\vec{BC}\) are parallel to each other, then we can easily show that A, B and C are collinear.

Therefore, \(\vec{AB}\) is given by

\(\vec{AB}\) = Position vector of B - Position vector of A

Let us note the relation between \(\vec{AB}\) and \(\vec{BC}\).

We know,

This relation shows that \(\vec{AB}\) and \(\vec{BC}\) are parallel to each other.

But also, \(\vec B\) is the common vector in \(\vec{AB}\) and \(\vec{BC}\).

⇒ \(\vec{AB}\) and \(\vec{BC}\) are not parallel but lies on a straight line.

Thus, A, B, and C are collinear.