VECTORS

http://www.physics.uoguelph.ca/tutorials/vectors/vectors.html

 

 

The reason for this introduction to vectors is that many concepts in science, for example, displacement, velocity, force, acceleration, have a size or magnitude, but also they have associated with them the idea of a direction. And it is obviously more convenient to represent both quantities by just one symbol. That is the vector.

 

Graphically, a vector is represented by an arrow, defining the direction, and the length of the arrow defines the vector's magnitude..

If we denote one end of the arrow by the origin O and the tip of the arrow by Q. Then the vector may be represented algebraically by OQ.

This is often simplified to just  Q or Q.

The line and arrow above the Q are there to indicate that the symbol represents a vector. Another notation is boldface type as:Q.

 

The magnitude of a vector is denoted by absolute value signs around the vector symbol: magnitude of Q = |Q|.

The operation of addition, subtraction and multiplication of ordinary algebra can be extended to vectors with some new definitions and a few new rules. There are two fundamental definitions.

                1) Two vectors,A and B are equal if they have the same magnitude and direction, regardless of whether they have the same initial points. 

                2) A vector having the same magnitude as A but in the opposite direction to A is denoted by -A ,

 

 

 

 

ADDITION


The sum of two vectors, A and B, is a vector C, which is obtained by placing the initial point of B on the final point of A, and then drawing a line from the initial point of A to the final point of B .

 

The operation of vector addition as described here can be written as C = A + B

 

SUBTRACTION

Vector subtraction is defined in the following way. The difference of two vectors, A - B , is a vector C , C = A - B
or C = A + (-B).Thus vector subtraction can be represented as a vector addition.


The graphical representation is shown below. Inspection of the graphical representation shows that we place the initial point of the vector -B on the final point the vector A , and then draw a line from the initial point of A to the final point of -B.

 

MULTIPLICATION BY A SCALAR

Any quantity which has a magnitude but no direction associated with it is called a "scalar". For example, speed, mass and temperature.

The product of a scalar, m say, times a vector A , is another vector, B, where B has the same direction as A but
|B| = m|A|.

UNIT VECTOR

Vectors can be related to the basic coordinate systems which we use by the introduction of what we call unit vectors.

A unit vector is one which has a magnitude of 1 and is often indicated by putting a hat on top of the vector symbol, for example UNIT VECTOR = a is read as "a hat" or "a unit".

 

COMPONENTS OF AVECTOR

Let us consider the two-dimensional Cartesian Coordinate System. We can define a unit vector in the x-direction by i; Similarly in the y-direction we use j. Any two-dimensional vector can now be represented by employing multiples of the unit vectors,

 

 

 

The vector A can be represented algebraically by A = Ax + Ay. Where Ax and Ay are vectors in the x and y directions. If Ax and Ay are the magnitudes of Ax and Ay, then Axi and Ayj are the vector components of A in the x and y directions respectively

 

 

RESOLVING A VECTOR

The breaking up of a vector into its component parts is known as resolving a vector. Notice that the representation of A by it's components is not unique. Depending on the orientation of the coordinate system with respect to the vector in question, it is possible to have more than one set of components.

It is perhaps easier to understand this by having a look at an example.

Consider an object of mass, M, placed on a smooth inclined plane, as shown in Panel 10 . The gravitational force acting on the object is
F = mg where g is the acceleration due to gravity.

 

In the unprimed coordinate system, the vector F can be written as

F= - Fy.j

but in the primed coordinate system : F= -Fxi - Fyj

Which representation to use will depend on the particular problem that you are faced with.

 

MULTIPLICATION OF VECTORS

The multiplication of two vectors, is not uniquely defined, in the sense that there is a question as to whether the product will be a vector or not. For this reason there are two types of vector multiplication.

- First, the scalar or dot product of two vectors, which results in a scalar.

- And secondly, the vector or cross product of two vectors, which results in a vector.

The scalar product of two vectors, A and B denoted by A·B, is defined as the product of the magnitudes of the vectors times the cosine of the angle between them.

Note that the result of a dot product is a scalar, not a vector.


The definition of the scalar product given earlier, required a knowledge of the magnitude of A and B , as well as the angle of the two vectors. If we are given the vectors in terms of a Cartesian representation, that is, in terms of i  and j we can use the information to work out the scalar product, without having to determine the angle between the vectors.

Check, by your own, the next web sites:

http://www.physics.uoguelph.ca/applets/Intro_physics/Vector_Add/vectortest.htm

http://www.pa.uky.edu/~phy211/VecArith/index.html

http://www.glenbrook.k12.il.us/gbssci/phys/Class/vectors/u3l1b.html

 

MORE ABOUT VECTORS

ACTIVITIES

Determine the magnitude and direction of the following vectors . Use the indicated scale and a scale conversion to determine the magnitude.

1. Given the SCALE: 1 cm = 10 m/s, determine the magnitude and direction of this vector.

 

 

 

 

2. Given the SCALE: 1 cm = 50 km/hr, determine the magnitude and direction of this vector.

 

 

 

 

 

 

 

3. Given the SCALE: 1 cm = 10 m/s, determine the magnitude and direction of this vector.

 

 

 

 

 

4. Given the SCALE: 1 cm = 50 km/hr, determine the magnitude and direction of this vector.

 

 

 

 

 

 

 

 

 

 

 

Do the next exercises:

1. Add the following vectors and determine the resultant: 3.0 m/s, 45 deg and 5.0 m/s, 135 deg

 

2. Add the following vectors and determine the resultant.: 5.0 m/s, 45 deg and 2.0 m/s, 180 deg

 

3. Add the following vectors and determine the resultant.: 6.0 m/s, 225 deg and 2.0 m/s, 90 deg

 

4. Add the following vectors and determine the resultant: 4.0 m/s, 135 deg and 4.0 m/s, 315 deg

 

5. Add the following vectors and determine the resultant.: 5.0 m/s, 45 deg and 2.5 m/s, 135 deg

 

6. Add the following vectors and determine the resultant.: 7.0 m/s, 0 deg and 2.0 m/s, 90 deg

 

7. Add the following vectors and determine the resultant.: 8.0 m/s, 330 deg and 4.0 m/s, 45 deg

 

8. Add the following vectors and determine the resultant.: 2.0 m/s, 150 deg and 4.0 m/s, 225 deg

 

9. Add the following vectors and determine the resultant.: 3.0 m/s, 45 deg and 5.0 m/s, 135 deg and 2.0 m/s, 60 deg

 

10. Add the following vectors and determine the resultant.: 2.0 m/s, 315 deg and 5.0 m/s, 180 deg and 2.0 m/s, 60 deg

 

11. Add the following vectors and determine the resultant.: 4.0 m/s, 90 deg and 2.0 m/s, 0 deg and 2.0 m/s, 210 deg

 

12. Add the following vectors and determine the resultant.: 2.5 m/s, 45 deg and 5.0 m/s, 270 deg and 5.0 m/s, 330 deg