vectors

[candice]

Hey

I am doing a presentation on vectors in maths class, and in my presentation I have to speak about how vectors (the sum and difference of vectors; and the product of a vector and a vector; and the product of a scalar and a vector) are applicable in life.

I also have to speak on how to determine the product of a vector and a vector and how to determine the product of a vector and a scalar.

And what is the norm of a vector? And how is it calculated?

I have done loads of research, but it is all extremely complex as it is all univercity mathematics and I am looking for something understandible to myself and my classmates in school.

Please please please help me to answer these questions in a clear and understandible way.

I have studied basic vectors in physical science, but that only really included the sum of vectors (the parallelogram method, the head to tail method etc. and drawing vector diagrams and then using triganometry to solve)

Please help - I really don't know where else to look

Thank you so much

[EDS]

Hi Candice,

Your best bet for a straightforward introduction to vectors may be a highschool or introductory university physics textbook. It might be worth asking the physics teacher at your school if he or she has any good books which treat the subject. (You may well end up receiving a personalized lecture as well!) A university mathematics treatment can be elegant and very general, but it's a harsh way to try to learn the basics, as you've discovered. Outside of that, browse through some books in the school or public library, of at a college library if there's one nearby.

You also might want to try a few web searches - there are probably some tutorials or published lectures online. Try adding the words "physics" or "astronomy" to the searches to weed out some of the detailed math treatments. One treatment that isn't necessarily the best, but is free and available for download, is available here: http://www.lightandmatter.com/area1book1.html#download (The link to the actual book is buried in the text. It's labeled "right-click here to download.")

A decent way to start is to read about how vectors can be represented in a coordinate system, if you haven't seen that already. Almost any treatment of vector math will require that you know how to write down a vector using it's components in x,y,z space. Try "vector decomposition" or some combination of "representation" "vector" and "coordinate system" as search keywords.

Hey
I am doing a presentation on vectors in maths class, and in my presentation I have to speak about how vectors (the sum and difference of vectors; and the product of a vector and a vector; and the product of a scalar and a vector) are applicable in life.

I suspect as you learn about the subject, examples will jump out at you. If you really get stuck, I'll be happy to offer some suggestions.

I also have to speak on how to determine the product of a vector and a vector and how to determine the product of a vector and a scalar.

The product of a vector and a scalar is pretty straightforward. If a vector is an arrow in space, then multiplying it by a scalar just means multiplying its length by the scalar and keeping its direction the same.

There are two different kinds of vector-vector products. The first results in a scalar, and is called a "dot product" or an "inner product." You can think of it as multiplying the length of the two vectors times the amount they overlap. (eg. if they point in same direction, it's just the two lengths multiplied together. If they point at 90 degrees to each other, then it's zero.) Google can give you the details of how to calculate the dot product in practice.

The second results in another vector, and it's usually called a "cross product" or a "vector product". It's a little less obvious why it is a useful quantity, but you can understand qualitatively by building a parallelogram with sides made out of the two vectors. The cross product has a length proportional to the area of that parallelogram, and a direction coming straight up out of the plane.

There are other even more abstract types of products that people define, but these are the basic ones. A few google searches with the keywords above should give you a start.

And what is the norm of a vector? And how is it calculated?

If your vector is an arrow in space, then the norm is just the length of that arrow. In Cartesian coordinates (the usual x,y,z rectangular coordinates), the norm is given by adding the individual components in quadrature. ( dl^2 = dx^2 + dy^2 + dz^2, where dl is the norm, dx is x component of the vector, and "^2" means squared.)

Good luck,
Erik

[candice]

Hey Erik

Thank you very much for your reply, it did help me... but please help me further! I'm sorry, I must seem rather slow to you, but since I have never worked with the z axis in the cartesian plane, but have only dealt with the x and y axi, the concept is a little difficult for me to grasp.

This is what I have done so far:


WHAT IS A VECTOR?

A vector is any physical quantity for which both magnitude and direction must be specified i.e. when asked for a vector; a number unit and direction are required. Vectors are often used to determine the position of objects in relation to each other. In order to understand what this means let us consider a physical quantity with which we are familiar i.e. distance.
The definition of distance is path length covered and there is no directional quantity associated with it. This is thus a scalar quantity) scalar quantities have magnitude only.) The vector quantity of distance is displacement. The definition of displacement is the shortest distance from the starting point to the finishing point (distance and direction from original to final position).

e.g. a man starts at the hall and runs to the grade 12 classroom and back. The distance covered is great, but the displacement is 0 as he starts and ends at the same point.

Examples of other vector quantities include - force, velocity, acceleration, momentum and electric field strength.

HOW ARE VECTORS REPRESENTED?

Vectors are represented on paper by lines, with arrowheads. The lines should be drawn to scale (to represent the magnitude of the quantity in reality), and the arrowhead show the direction.

A rough sketch need not be to scale, but must be carefully labelled so that the magnitude and direction of the quantity are clearly shown.

When writing of a vector we generally write a capital letter with an arrow over it. The length of a vector (norm) is represented using the absolute value sign.







WHAT IS MEANT BY THE NORM OF A VECTOR?

The norm of a vector is another way of saying the length or magnitude of a vector.

How is it calculated?

To find the norm of a vector, one must work in more than one plane (i.e. take a dimensional look at the problem) – using 3 sets of axis. Lets call them x, y, and z.

Thus when working with the right angled triangles we cannot simply use Pythagoras’ theorem, but must use the three dimensional form of his theorem v = x2+ y2 + z2
i.e. the square root of the sum of the squares on its vector components.


THE SUM OF VECTORS

When working with vectors, one should always start off by drawing a rough sketch of the quantities involved. To find the sum of vectors, one needs to find the single vector that could be used to replace all the vector components – one vector that will have the same affect as all the other vectors separately (i.e. the resultant)

There are numerous methods used to find the sum of vectors, and it is using a combination of these methods that allows us to get to the required answer.

· Algebraic sum – the same of vectors along the same straight line is found by simply adding their magnitudes, but since direction is involved, some magnitudes will be negative and others positive (depending on in which direction they are pointing).
e.g. (+2) + (-4) = -2 therefore the resultant is 2 units in the negative direction.









· Head to tail method – in order to use this method, the vectors must also be along a straight line (i.e. along the x and y axis of the Cartesian plane). The vectors must be arranged on the Cartesian plane so that the head of one arrow touches the tail of the next and this should be done in a clockwise or anticlockwise order. (It is very important that the magnitude and direction is not altered in the ‘re-arrangement’ process.) One all the vectors are placed head to tail, a last arrow (the resultant) must be drawn from point of origin to the ending point (i.e. from A to E)








Because the head to tail method requires the vectors to be parallel to the x and y-axis, we have to find a way to convert the vectors that lie at an angle between the axis into those we can work with. We do this by converting these vectors into their components.

To do this we must draw a rough diagram of the force in question. Its ‘rectangular’ components will always result in a right-angled triangle being formed (with the force in question and its 2 components forming the 3 sides). When dividing into components, the diagram on the Cartesian plane should not be drawn head to tail, but rather the tail of the original vector should join with the tail of its horizontal component.







In triangle ABC
Cos 2 = adjacent
Hypotenuse
Cos 2 = AC
AB
AB = AC
Cos 2

In the same way the vertical components can be found using trigonometry.

Sin 2 = opposite
Hypotenuse
Sin 2 = BC
AB
AB = BC
Sin 2

This must be done to all the vectors that are not || to the x or y-axis.

Once all the ‘rectangular’ components have been found, we can disregard the original (non-rectangular) vectors and use the components.

We can thus add all the vertical components together using the algebraic method and in the same way add all the horizontal components together.

We can then draw our two resultant components together on a Cartesian plane and finally we can change them out of their component form and into one vector by using the head to tail method.

An example of where all this would have to be done is in a question such as:

QUESTION
Find the resultant of a force of 12N exerted on a bearing of 45 and a force of 5N on a bearing of 180

ANSWER

Step 1 – split the vectors into their components








Cos 45 = adjacent Sin 45 = opposite
Hypotenuse hypotenuse
Cos 45 = adjacent Sin 45 = opposite
12 12
12Cos45 = horizontal component 8,49N 12Sin45 = vertical component 8,49N










Step 2 – Find the sum of these components using ‘algebraic sum’

Vertical = (+8,49) + -5) = 3,49N
Horizontal = (+8,49) + 0 = 8,49N

Step 3 – Redraw these new overall components on the Cartesian pane and commence with the head to tail method








We can use Pythagoras’ theorem (or trigonometry) to find the magnitude of the resultant.

(8,49) + (3,49) = AB
AB = 9,18N

And the angle through which the vector moves (need for directional purposes) can be determined using trigonometry.

Tan 2 = opposite
Adjacent
Tan 2 = 3,49
8,49
2 = 22,35



Step 4 – decipher the answer

It is best to give direction by saying “on a bearing of�, and thus we should say (90 –22,35)
And thus our answer would be 9,18N on a bearing of 67,65
PARALLELOGRAM METHOD

This method is used to find the resultant of vectors acting on the same point and can only be used to add 2 vectors.

Begin by sketching the two vectors ‘on the Cartesian plane (as the two sides of a parallelogram).








Complete the parallelogram remembering that opposite sides are equal and parallel.








The resultant is the diagonal running from where the tails of the two original vectors meet to the opposite corner.

To determine this algebraically, we use our sketch as reference.

Since AB = 10
CD = 10 (opposite sides of a parallelogram=)
Since AC = 4
BD = 4 (opposite sides of a parallelogram=)

Therefore we can calculate using either )ABD or ) ACD as we have two of the side lengths for both.

In )ABD
AD2 = AB2 + BD2 (Pythagoras)
AD2 = 10 + 4
AD = 116

We use trigonometry to find direction (angles etc.)
Tan 2 = opposite
Hypotenuse
Tan 2 = 4
10
2 = 22,8

Thus the resultant vector is 116 units on a bearing of (90 –22,8 ) = 67,2
DIFFERENCE OF VECTORS

When subtracting vectors split them into their rectangular component and then find the difference of the vertical and horizontal components separately and then use the head to tail method to find the resultant.

Step 1 – Split the vectors into their vertical and horizontal components









Sin 2 = opposite Sin2 = opposite
Hypotenuse hypotenuse
Sin30 = opposite Sin2 = opposite
9 7
9Sin30 = vertical component 4,5 7Sin45 = vertical component 4,95

Cos 2 = adjacent Cos2 = adjacent
Hypotenuse hypotenuse
Cos 30=adjacent Cos45 = adjacent
9 7
9Cos30 = horizontal component 7,79 7Cos45 = horizontal component 4,95

Step 2 – Find the algebraic difference

Vertical components: (+4,5) – (+4,95) = -0,45
Horizontal components: (+7,79) – (-4,95) = +12,74

Step 3 – Redraw these new components on a Cartesian plane and then commence with the head to tail method



















(0,45)2 + (12,74) 2 = AC2 (Pythagoras)
AC = 12,75 units

tan 2 = opposite
adjacent
tan 2 = 12,74
0,45
2 = 87,98

Step 4 – decipher the answer

(90 – 87,98) + 90 = 92,02

Therefore the resultant of the subtracted vectors is 12,75 units on a bearing of 92,02

THE PRODUCT OF A VECTOR AND A SCALAR

A vector multiplied by a scalar is a vector. To understand this let us consider the example:
4 m west multiplied by 3 = 12 m west

Or consider - If a car is moving in a northerly direction and it suddenly duplicates (becomes 2 cars) those two cars will still be travelling in a northerly direction.

THE PRODUCT OF A VECTOR AND A VECTOR

There are 2 methods of finding the product of 2 vectors, namely: the scalar product method and the vector product method.

To find the scalar product the formula used a b = a b cos t (The product of the magnitudes of the vectors multiplied by the cosine of the angle between them). Where a and b are the two vector quantities and t is the distance between them.

Normally the product of two vectors is a scalar quantity but if one wants to acquire a vector product then one has to solve the equation taking three dimensions into account i.e. to find the vector product of the vector quantities a and b one uses the equation where n is the vector perpendicular to the plane in which a and b lie, and t is the angle between a and b.

An example of an application of the scalar product is the analysis of movement.


The spaces indicate places where I am to draw diagrams and i apologise for the working out being all jumbled (you might want to ignre those).

But you see, I don't even know if what I have written makes sense - does it? Is it mathematically correct?

I am on holiday at the moment, but I am going back to school tomorrow so I will speak to my science teacher as you suggested.

Please can you try to explain to me the three planes (x,y,z) and how they affect vectors. This is not one of the questions I have to answer in my presentation, but I'm sure it would help me to understand.

Thank you again

[candice]

Hey everyone1

Please can you try and help me with this! My presentation is looming and I really don't want to stand up and talk about something I don't really understand!

What are the applications of vectors in life?

What is the norm of a vector?

How do you multiply a vector and a vector?

How do you multiply a scalar and a vector?

Please help me and try and do so in as simple wording as possible!
I would really really appreciate it

Thanks
Candice

[EDS]

Hi Candice,

I'm sorry it took so long to get back to you. It's been an outrageously busy week and I haven't stopped by the board for a bit.

On the whole, your writeup looks good. I obviously don't know what your teacher is expecting, so weight that statement appropriately.

Also, you shouldn't worry about "seeming slow." It's really hard to tell what someone alreay knows or doesn't from a message board post, so it's easy to assume background that isn't there. Everyone starts learning this stuff somewhere, and there's no shame in being new to it. You, and everyone else in this forum, should feel absolutely free to say, "Hold on, I didn't understand that" whenever it's appropriate.

A few detailed comments:

1 - When calculating things about vectors (like the norm), you can work in as many dimensions as you're interested in. If you're only dealing with a 2-dimensional plane, then the plain old Pythagorean theorem is fine.

2 - The "head to tail" method should work for ANY vectors, not just those which lie along the X or Y axis. I'm not sure I understand the part about things being clockwise or anti-clockwise. You might want to try a few google searches for some examples to make sure you understand that part.

3 - Breaking vectors into x,y components and adding components is equivalent to the head-to-tail method. Think of it as the algebraic way of doing the same thing you do with geometry.

4 - Writing "a force of 12N exerted on a bearing of 45" seems a bit strange to me. That isn't to say it's wrong - just something I've never seen before. (A more usual way to write it would probably be something like "a force of 12 N applied at an angle of 45 degrees to the surface.")

5 - Is there a way to find the difference of vectors using something like the head-to-tail method?

6 - Examples of vector-products aren't too usual in every day life. Check out "angular momentum" or "magnetic field" in the index of a high-school physics textbook for examples. If you're class is only interested in working in a two dimensional XY plane, they may not be interested in vector products at all.

7- Most of the math one can do in three dimensions looks almost the same as in two dimensions. If you understand how vectors work in 2D, then you've got the basics. There are a couple extra details you have to keep track of, but the basic idea is the same.

Best,
Erik

[shijun]

Hi Candice,

I see that you've received some good advice so far from the Science Buddies Experts. Please note that for beginners, learning about vectors may not appear straighforward, but if you are patient and willing to spend some time thinking about them, it will start to make sense. Feel free to do some internet research on your own and read some additional materials. As with math and physics, it's impossible for anyone to simply feed you the answer. A good intuitition comes with practice and experience, and you should try your best to spend some time with it before throwing in the towel. It's okay to ask for help, but make sure you've actually spent some time thinking about your questions beforehand. The learning experience will be much more rewarding!

[candice]

Thanx guys

I'm still lookin for sum answers but thanks a lot for all your help so far -ED especially!
Hopefully I will be able to get hold of some text books to help explain - cause the internet sites I have found just aren't giving me what I'm looking for!

Will let you know if I make any 'understanding discoveries'

Thanx again

Oh - ED - you asked if there was a way to find the difference of vectors using something like the head-to-tail method... To be honest I just thought it was logical and made up how using how it is done to find the sum - is it wrong??? OOPS if it is!

[EDS]

Hi Candice,

Oh - ED - you asked if there was a way to find the difference of vectors using something like the head-to-tail method... To be honest I just thought it was logical and made up how using how it is done to find the sum - is it wrong??? OOPS if it is!

Nope - it's not wrong at all. I was just suggesting that since you talk about finding the sum of vectors using both methods but you only talk about finding the difference using the algebraic method, you might want to think about whether you could also calculate a difference in a head-to-tail sort of way. (Consider subtracting as "adding a negative.")

Best,
Erik