VECTORS AND
TRANSFORMATIONS
By Thanav Suriyakumar
9-C
What are Vectors ?:
A vector is a type of quantity generally used in physics and mathematics. Whereas
a scalar is a quantity that only has magnitude, a vector has magnitude and
direction. Vectors are used in a variety of fields in our day to day life. Vectors are
represented in the form of a column vector which is similar to a coordinate but
contains the x value above
the y value. The x value
represents the displacement
of the final position of the
value from its original
position along the x axis
and the y value is the same
but along the y axis. For
example the vector to the
right has a displacement of
3 on the x axis and 1 on the y axis. Using a column vector, vectors can be mapped
on to a diagram. An arrow is shown on the diagram to represent its direction.
Vector Operations:
Operations such as addition, subtraction, and multiplication
can be performed with vectors. Vector addition is similar to
normal addition when done with column vectors. A vector
of (4 , 2) added to (-1, 3) gives a
vector of (3, 5). Vector diagrams can
also be added together by connecting
both vectors based on their directions.
A vector can be turned into it’s negative by reversing it’s
sign. Subtraction of vectors works in the same way as
normal subtraction.When a vector of (3, -1) is subtracted
from (5, 8) it gives a vector of ( 2, 9). The negative of the subtracted vector is
drawn and connected to the other vector when subtracting with vector diagrams.
When a vector of (4, 6) is multiplied by 2 it turns into a vector of (8, 12)
Magnitude of Vectors:
The magnitude or length of a vector can be
determined using the pythagorean theorem. This
means that the root of the sum of the square of the x
and y value of the vector gives its magnitude. This
is because the x and y lengths form a right angled
triangle with the vector as the hypotenuse.
This is shown by the image on the left.
Transformations:
A transformation in terms of vectors is an application that changes a vector’s
length, orientation, position, or all three. There are many types of transformations
which each change vectors in different ways
Some examples include:
1. Translation
2. Stretch
3. Enlargement
4. Rotation
5. Reflection
Translation:
A translation is a form of transformation which moves
the vector by a certain amount on a grid. Translations
only change the position of a vector. Translations can
be horizontal, vertical, or both. The amount by which a
vector is translated can be written as a column vector,
this can be measured by counting the movement on the
x and y axis. For example, the figure to the right is
translated by a column vector of (4 , -9). Adding the
translation amount in it’s column vector form to a
vector will give the column vector of the translated
vector.
Stretch:
A stretch is a form of translation where
the length of one axis of a figure of
vectors is changed by a certain factor. The
position of the vector from an axis also
changes by the factor based on it’s
original distance from the axis. The vector
is also parallel to a line which is known as
the invariant line incase it is specified.
Triangle 1 has been transformed into
triangle 3 by stretch with factor 3 and an
invariant x axis. Triangle 1 has been
transformed into triangle 2 by stretch factor 3 with an invariant y axis. Triangle 1
has been transformed into triangle 3 by stretching both axes by factor 3
Enlargement:
Enlargement is when a figure’s size is changed by a certain factor from a specified
center of enlargement. This changes a figure’s length and position. The factor by
which the figure is enlarged is known as the scale factor, all
lengths are multiplied by this factor. These lengths are drawn
around the center of enlargement which is the center of the
enlarged figure. The figure to the left has been enlarged by scale
factor 2 about the point B. The center of enlargement for this
figure is around the point (4,7), this point cannot be accurately
determined. The scale factor of a figure can be found by dividing
the enlarged length by the original length.
Rotation:
A rotation is when a figure’s direction is changed by a
certain angle, usually by multiples of 90. This rotation can
be clockwise or anticlockwise. Generally the direction
which gives a lower angle of rotation is taken when
describing a rotation transformation. The angle of rotation
can be found by looking at the image. A protractor can be
used to draw an image about a certain angle. This all done
about a center of rotation. The figure to the left shows a
rotation of 180 (clockwise or anticlockwise) about the
center of rotation (0,0).
Reflection:
A reflection is a transformation in which is
reflected/flipped like it would be in a mirror across a
mirror line. The mirror line is generally in between
the figure and its image. The image to the left is
reflected about the mirror line whose equation is y =
x. The closest point of the figure and image to the
mirror line are both the same distance from the
mirror line. Several transformations may be combined together to give a final
figure which is completely different from the original figure.
Real Life Uses of Vectors:
Vectors are used in a variety of different fields
1. Nautical Movement
For example, when a boat is crossing a river the boat may be going against the
current. In order to calculate the resultant speed of the boat a diagram can be drawn
with both speeds, the angle of travel of the boat can also be found
2. Military
The military can use vectors to calculate traveling time and location when using
ships and tanks. They can be used to find a route which bypasses enemy defenses.
Artillery uses vectors to calculate the angle at which a projectile may be optimally
shot in order to hit a target or maximize damage
3. GPS
Satellite signals can be triangulated in order to calculate the location of an object
using vectors, radars may also use similar techniques. Air traffic control uses
vectors to guide aircraft without them crashing
Conclusion:
Vectors are quantities with magnitude and direction. Vectors can be drawn and
represented in a variety of ways. They can be added, subtracted, or multiplied.
A transformation is when a property of a vector is changed. They include
translations, stretch, enlargement, rotation, and reflection. Vectors are used in many
crucial areas such as in the military.