Hello friends, today I will talk about coplanar vectors. Have a look!!
First of all, I just explain the phrase ‘coplanar vectors’.
When all the vectors are in the same plane, they are called coplanar vectors. So that means, if the vectors are not in the same plane, they can’t be coplanar.
That’s easy enough, I think.
Now the question is how to determine whether some vectors are coplanar or not.
How to find out coplanar vectors
One of the most common methods is to use the scalar triple product of vectors.
If there are three vectors and the scalar triple product of these vectors will be
The value of is the volume of a parallelopiped with vectors and as its adjacent sides.
Now when all these vectors will be in one plane, then the parallelopiped can not be formed. That means the volume of the parallelopiped will be zero.
In other words, coplanar vectors ⇒ volume of the parallelopiped = 0.
Also the volume of the parallelopiped = magnitude of the scalar triple product of vectors.
This means coplanar vectors ⇒ the scalar triple product of vectors = 0.
Now I will solve some examples of coplanar vectors.
Examples of coplanar vectors
Note: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.
So here is the first example.
According to Stroud and Booth (2011)* “Determine whether the three vectors are coplanar.”
Here the given three vectors are:
Now any three vectors will be coplanar, if the scalar triple product of them is zero.
So my next job is to find out the scalar triple product of these three vectors.
Here I will find out the scalar triple product
Thus it will be
As a next step, now I’ll evaluate the determinant on the right-hand side to get the value of
Thus it will be
Now I can see that the scalar triple product equals to zero. Hence I can conclude that these three vectors are coplanar.
This is the answer to this example.
Now I will move to the next one.
According to Stroud and Booth (2011)* “Determine the value of such that the three vectors are coplanar when .”
In this example, the given vectors are
Now I have to determine the value of so that these three vectors will be coplanar.
As I have already mentioned earlier, for coplanar vectors, the scalar triple product will be zero.
Thus the scalar triple product will be
Now I’ll evaluate the determinant on the right-hand side to get the value of as
I already know that these three vectors and are coplanar.
Therefore the scalar triple product of these vectors equals to zero.
This means equals to zero. Now I’ll solve it to get the value of . Thus I get
Thus I can conclude that for the three vectors are coplanar when .
So here ends my second as well as the last example.
Dear friends, this is the end of my today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!