Hello friends, today I will talk about the scalar and vector products of two vectors.

### Scalar and vector products of two vectors

Suppose there are two vectors and .

Now also let me assume and

The scalar product of the vectors and is

Similarly, the vector product of the two vectors and is

So I can also say that

- scalar product of two vectors is a scalar.

- vector product of two vectors is a vector.

Also, these scalar and vector products of two vectors are used a lot in vector analysis.

For example, the scalar product of two vectors is used to find out the directional derivative of a surface.

In the same way, both the scalar and vector products of two vectors are used to check the coplanarity of vectors.

And of course, these are only a few examples. There are many more uses as well like the scalar triple product, vector triple product and so on.

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.

##### Example 1

According to Stroud and Booth (2013)* “Find the scalar product and the vector product when and ”

##### Solution

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.

##### Example 2

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 ”

Solution

Here the given vectors are and

I’ll start with part (a) of the example.

###### (a)

According to the formula for scalar product of two vectors and will be

Now also means Here is the angle between two vectors and

In this example, equals to zero. 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).

###### (b)

Here, 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

Now 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!!

nganga says

These are very clear steps that can be easily followed