Differentiation of vectors. Hello friends, today it’s all about differentiation of vectors. Have a look!!

### Differentiation of vectors

‘Differentiation of vectors’ is a quite simple straightforward thing.

It happens when the vector has a parametric form like .

Also, the differentiation of vectors follows standard rules of differentiation.

###### Where to use ‘differentiation of vectors’

Differentiation of vectors is used to get the equation of unit tangent vector in vector analysis.

Just have a look at these two examples!!

#### Examples of differentiation of vectors

Disclaimer: 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 (2011)* “If and , determine (a) (b) (c) ”

##### Solution

Now here I have two vectors, and

Also I know that the vector is . Also the vector is

So I will start with part (a).

###### (a)

As I can see, I have to get the value of

Now means the scalar product of two vectors and

As per the formula of scalar product of two vectors, the value of is

Now I’ll simplify it to get

Thus I get the value of as

(1)

Now I will differentiate both sides of equation (1) with respect to For that, I have to use the standard formulas for differentiation.

So I get

And this means

Next, I’ll simplify it to get

Hence I can conclude that is the answer to part (a).

Now I move to part (b).

###### (b)

Here I have to get the value of

Now means the vector product of two vectors and

As per the formula of vector product of two vectors, the value of is

So this gives

Next I will simplify it to get

Thus I get the value of as

(2)

Again I will use standard formulas in differentiation to differentiate both sides of equation (2) with respect to

Therefore it will be

So this gives

Hence I can conclude that is the answer to part (b).

Now I move to part (c).

###### (c)

Here I have to get the value of

Now means adding two vectors and

Thus it will be

Now I’ll simplify it to get

Next, I will differentiate both sides with respect to to get

Hence I can conclude that is the answer to part (c).

This is the end of example 1. Now I will work on the second example.

##### Example 2

According to Stroud and Booth (2011)* “If determine (a) (b) (c) the value of at .”

##### Solution

Now here the given vector is So I will start with part (a) of the problem.

###### (a)

Now I have to get the value of

Like the first example, here also I will differentiate the vector following standard rules of differentiations.

Thus it will be

Next, I’ll simplify it to get

This is the answer to part (a). Now I’ll move to part (b).

###### (b)

Next, I have to get the value of

Now, I have already got the value of from part (a).

Thus I will differentiate both sides of with respect to to get the value of as

So it will be

Hence I can conclude that is the answer to part (b) of this example.

Next, I will solve the next part (c).

###### (c)

Now I have to get the value of at .

First of all, I will get the value of

Now, means the absolute value of

Suppose is a vector with

Then the absolute value of the vector will be

So, in this example, the value of will be

Hence it gives

Now at it will be

Also, I already know that and .

Hence I can say that

and

Thus the value of will be

Thus my conclusion is that the value of at is

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!!

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