Hello friends, today I will talk about the scalar and vector products of two vectors.
Looking for more in vectors?? Do check out:
Now the question is: where to use the scalar and vector products?? For example,
- The scalar product of two vectors is used to find out the directional derivative of a surface.
- Both the scalar and vector products of two vectors are used to check the coplanarity of vectors.
[pdf-embedder url=”https://www.engineeringmathgeek.com/wp-content/uploads/2019/09/Scalar-and-vector-products_compressed.pdf” title=”Scalar and vector products_compressed”]
Scalar and vector products of two vectors
Suppose there are two vectors and .
Now also let me assume and
So the scalar product of the vectors and is
Similarly, the vector product of the two vectors and is
Thus I can also say that
- scalar product of two vectors is a scalar.
- vector product of two vectors is a vector.
Now I will solve some examples of scalar and vector products of two vectors.
Examples of the scalar and vector products of two 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 (2013)*, “Find the scalar product and the vector product when and ”
Here the given vectors are and
Thus as per the formula for scalar products of two vectors, the scalar product will be
Now I’ll simplify it.
So it will be
In the same way, now I will get the vector product of two vectors and .
Therefore according to the formula for vector products of two vectors, the vector product will be
Now I will expand the determinant on the right hand side. Thus it will be
Next I will simplify it further to get
Thus I can conclude that the scalar product is 8. Also the vector product is
These are the answers to this example.
Now I’ll go to the next example.
According to Stroud and Booth (2013)*, “ and are vectors defined by and where and are mutually perpendicular unit vectors.
(a) Calculate and show that and are perpendicular to each other.
(b)Find the magnitude and the direction cosines of the product vector ”
Here the given vectors are and
I’ll start with part (a) of the example.
According to the formula for scalar product of two vectors and will be
Next, I’ll simplify it to get
Now also means Here is the angle between two vectors and
And in this example, equals to zero. So that means
As I already know, for Thus I can say, in this case also .
That means the angle between vectors and is I can also say that, the vectors and are perpendicular to each other.
So part (a) of the example is proved.
Now I’ll do part (b).
First of all, I’ll find out the vector product
As per the formula for vector product of two vectors, will be
Now I will expand the determinant on the right-hand side. Thus it will be
Next, I will simplify it further to get
Therefore the magnitude of the vector is
Thus will be
Thus the direction cosines of the product vector are
Hence I can conclude that 68.53 is the magnitude of the product vector Also, its direction cosines are
Now this is the answer to part (b) of this problem.
Dear friends, this is the end of 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!!