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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