The unit vector notation used in Engineering mathematics is wonderfully simple and powerful.

Imagine we have a position in the x/y plane, lets call it P, and we want to form a vector from the origin to this point. We call that position vector

**OP**.

Lets say that P is located at x=2 and y=3, now we can draw a line from the origin to P and put an arrow head along it. We have a vector.

Now lets define the unit vector (i.e. one unit in a particular direction) in the x direction as

**and the unit vector in the y direction as**

*i***.**

*j*We can now say OP = 2

*+*

**i***3*

**j.**Lets try a few things ....

What is the magnitude of this vector ?

If we draw a right angled triangle from

**OP**, we can quickly see that the magnitude of OP must be equal to the square root of (2 squared + 3 squared) = sqrt(13).

What is the unit vector of

**OP**?

It must be

**OP**/sqrt(13) i.e. one unit in the direction of

**OP**.

What about if we want to add another vector (

**OQ**= 4

**+ 1**

*i***) to**

*j***OP**?

**+**

OP

OP

**OQ**= (4 + 2)

**+ (3 + 1)**

*i***= 6**

*j***+**

*i***4**.

*j*What if I want to find the vector

**QO**?

**= -**

QO

QO

**OQ**= -(4

*+*

**i***1*) = -

**j****4**-1

*i*

*j*What about if we want to find the vector

**PQ**?

We use the head to tail rule and say

**PO**+

**OQ**=

**PQ**= -(2

**+ 3**

*i***) + (4**

*j***+ 1**

*i***) = 2**

*j***- 2**

*i*

*j.*
This approach makes the whole problem of manipulating vectors easy and more powerful than constantly referring to angles and scalar qauntities.

First comment. And yes, sir, now you may have your "sastifaction" and may laugh loudly in your wife's face [this blog has been made in 2009, so I would presume this would be a pretty big comeback for you]. And may I also ask for a favor in return, please try to maybe use pictures in your explainations/examples? When I read on the computer, I tend to skim through and I don't really picture the vector positions.

ReplyDeleteThanks, frank_walker007