Download Equality of Column Vectors

1. What is a Vector? Why Study Vectors?
http://www.onlinemathlearning.com/what-is-vector.html and http://www.intmath.com/Vectors/Vectors-intro.php
You use vectors in almost every activity you do. Examples of everyday activities that
involve vectors include:

Breathing (your diaphragm muscles exert a force that has a magnitude and
direction)

Walking (you walk at a velocity of around 6 km/h in the direction of the
bathroom)

Going to school (the bus has a length of about 20 m and is headed towards your
school)

Lunch (the displacement from your class room to the canteen is about 40 m in a
northerly direction)
Vector quantities are quantities that have both magnitudes and directions such
velocity (speed and direction), force and acceleration.
Examples of Vector Quantities:

I travel 30 km in a Northerly direction (magnitude is 30 km, direction is North this is a displacement vector)

The train is going 80 km/h towards Sydney (magnitude is 80 km/h, direction is
'towards Sydney' - it is a velocity vector)

The force on the bridge is 50 N acting downwards (the magnitude is 50 Newtons
and the direction is down - it is a force vector)
Other examples of vectors include: Acceleration, momentum, angular momentum,
magnetic and electric fields
Each of the examples above involves magnitude and direction.
Scalar quantities are quantities that have only magnitudes such as time, area and
distance.
A vector is not the same as a scalar. Scalars have magnitude only. For example, a
speed of 35 km/h is a scalar quantity, since no direction is given.
Other examples of scalar quantities are: Volume, density, temperature, mass, speed,
time, length, distance, work and energy.
Each of these quantities has magnitude only, and do not involve direction.
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2. Vector Representation
A vector is drawn using a directed line segment. The length of the arrow indicates the
magnitude of the vector. The direction of the vector is represented by the direction of
the arrow. Such a vector is called a "localized vector".
Some textbooks use a bold capital letter to name vectors. For example, a force vector
could be written as F. Some other textbooks write vectors using an arrow above the
vector name, like this:
The above vector can be denoted by
or AB or
and B is called the terminal point of
.
or
or a. A is called the initial point
You will also see vectors written using matrix-like notation. For example, the vector
2
acting from (0, 0) in the direction of the point (2, 3) can be written   . This is an
3
ordered pair called a column vector.
Consider the line PQ in the diagram. The line represents the
translation of P to Q, which is 2 right and 3 up.
This can be written as the
ordered pair
Example:
Express
as a column vector.
Solution:
The translation of C to D is 4 right and 3 down.
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3. Vector Magnitude
http://www.onlinemathlearning.com/vector-magnitude.html and
http://www.intmath.com/Vectors/Vectors-intro.php
The length of a vector is called the magnitude or modulus of the vector. The
magnitude of a vector is the length of the corresponding segment.
Example:
Express each of the following vectors as a column vector and find its magnitude.
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4. Equal Vectors
http://www.onlinemathlearning.com/equal-vectors.html and
http://www.intmath.com/Vectors/1_Vector-concepts.php
Vector OP has initial point O and terminal point P. The length of the line OP is an
indication of the magnitude of the vector.
We could have another vector RS as follows. It has initial point R and terminal point S.
Because the 2 vectors have the same magnitude and the same direction (they are both
horizontal and pointing to the right), then we say they are equal. We would write:
OP = RS
Equal vectors have the same magnitude and the same direction.
Equal vectors may start at different positions. Note that when the vectors are equal, the
directed line segments are parallel.
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Equality of Column Vectors
If two vectors are equal then their vector columns are equal.
Example:
The column vectors p and q are defined by
. Given that p = q
(a) find the values of x and y, (b) find the values of
Solution:
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and
, (c) express y in terms of x
5. Multiply Vector by a Scalar
http://www.onlinemathlearning.com/multiplication-vector.html and
http://www.intmath.com/Vectors/1_Vector-concepts.php
A scalar quantity has a magnitude but no direction. For example, a pen may have
length "10 cm". The length 10 cm is a scalar quantity - it has magnitude, but no
direction is involved.
In vectors, a fixed numeric value is called a scalar. We can increase or decrease the
magnitude of a vector by multiplying the vector by a scalar.
When vector x is multiplied by 3, the result is 3x.
When vector x is multiplied by –2, the result is –2x.
Example 1:
We have 3 weights tied to a beam. The first weight is W1 = 5
N, the second is W2 = 2 N and the third is W3 = 4 N.
We can represent these weights using a vector diagram
(where the length of the vector represents the magnitude)
as shown on the right:
They are vectors because they all have a direction (down)
and a magnitude.
Each of the following scalar multiples is true for this
situation:
W1 = 2.5 W2 , W2 = 0.5 W3 , W3 = 0.8 W1
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Example 2:
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6. Parallel and Collinear Vectors
http://www.onlinemathlearning.com/parallel-vectors.html
Vectors are parallel if they have the same direction. Both components of one vector
must be in the same ratio to the corresponding components of the parallel vector.
Example:
Collinear vectors are parallel and lie on the same straight line.
If AB and AD are parallel, then they are definitely collinear, since they share point A.
If AB and CD are collinear, then AB   AC   AD , where  and  are scalars.
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