Vectors

Vectors

Introduction
During a visual approach (the approach to a runway at an airport conducted under instrument flight rules), the Air Traffic Control instructs pilots to fly a particular heading (direction) for a certain distance (magnitude). This is exactly what a vector is; a quantity that has both magnitude and direction. This is why the Air Traffic Controllers might sometimes use the phrase “expect vectors for the visual approach…” when the plane nears the airport.
The real-life applications of vectors can be understood further by enriching your knowledge about vectors and translations which you studied in Senior Two. In this topic, you will be exposed to the nature of vector quantities, manipulation and re resentation of vectors in order to solve problems.

4.1 Describing Position Vectors Geometrically and as Column Vectors
In Senior Two, you learnt about a vector as a physical quantity that has both
magnitude and direction, and a translation as an example of a vector.
Activity 4.1 (a)
(a) Describe your journey from home to school and from school to home.
(b) Show the two journeys in a drawing.
(c) What conclusions do you make from the drawing you have made?

Describing position vectors geometrically
The translation of any point P(x, y) from the origin is called a position vector
Activity 4.1 (b) (Work in groups)
(a) Plot the points A(5, 2), B(-3, 4), C(-5, -2) and D(3, -4) on the same of pair axes.
(b) Join each point in (a) to the origin.
(c) Show the vector representation of each point from the origin on the graph.
(d) Read and write down the position vector of each point.
Describing position vectors as column vectors
From Activity 4.7(b), you wrote the position vectors of points. These can be written as column vectors.
Activity 4.1 (c) (Work in groups)
The positions ofAmongi and Nsibirwa in a classroom are (2, 4) and (3, 5), respectively.
(a) Represent their positions geometrically.
(b) Find the column vectors from Amongi to Nsibirwa, and from Nsibirwa to Amongi.
(c) Describe the relationship between the two column vectors you have obtained in (b) above.

4.2 Finding the Vector of a Directed Line Segment when Position Vectors of the Endpoints are Known
The vector of a directed line segment can be described as a column vector using the position vectors of the endpoints.

Activity 4.2(a) (Work in groups)

4.3 Finding the Position Vector of the Midpoint of a Line Segment Activity 4.3 (Work in groups)

4.3 Finding the Position Vector of the Midpoint of a Line Segment Activity 4.3 (Work in groups)

4.4 Using Vector Method to Divide a Line Proportionately,
Internally and Externally
Defining a directed line segment as a vector enables us to divide the line in-between and outside its end points.

Using vector method to divide a line internally Activity 4.4(a) (Work in groups)
Suggested materials:

• nails (pins)
• a thread
  Instructions:
• a piece of chalk
• soft boards
  (a) Fix the nails given to you on a surface in a straight line at equal intervals and label them.
  b)Tie the thread around all the nails, at once.
  c)How many parts of the thread have you formed?
  d)Name each part in terms of vectors.
  e)Write the parts of the thread in terms of ratios of their vectors.
  f)Write the ratio of each part of the thread to the whole thread.
  d)Identify one nail between the others and remove it. Explain your observation.

Using vector method to divide a line externally
Activity 4.4(c) (Work in groups)
(a) Plot points P(3, 2) and Q(ll, 6) on a graph paper.

b)Join P to Q using a straight line. Extend the line beyond point Q.
c)Find the horizontal displacement from P to Q and divide it into four equal parts.
d)Extend the horizontal displacement by a unit which is equivalent to one part in (c).
e)Identify the x-coordinate of point R.
f)Find the vertical displacement from P to Q and divide it into four equal parts.
D)Extend the vertical displacement by a unit which is equivalent to one part in (f).
e)Identify the y-coordinate of point R.
f)Write down the position vector of point R.
g)What do you conclude about point R?

4.5 Using Vectors to Show Parallelism
In this subtopic, you will learn about equal vectors and thus be able to use vectors
to show parallelism.
Illustrating equal vectors
Activity 4.5(a) (Work in groups)
Given the points E(3, 1), G(5, 2), 1(-6, -3) and -2);
(a) find all possible displacement vectors.
(b) what conclusion do you make about the displacement vectors you found?
(c) present your findings to the rest of the class.

Activity 4.5(b) (Work in groups)
Draw the coordinate grid of Activity 4.3 and answer the following questions:
(a) Identify the parallel line segments from the polygon formed.
(b) Find the position vectors of the endpoints of each pair of parallel segments in (a).
(c) Compare the position vectors above and give a comment.
(d) What do you conclude about the position vectors of the endpoints of parallel line segments?
Using scalar multiplication of a column vector to show parallelism

4.6 Using Vector Methods to Show Collinearity
In the previous subtopic, you learnt how to use vectors to show parallelism. Now, you will learn how to show collinearity using vectors.
Showing collinearity of column vectors
Activity 4.6(a) (Work in groups)
(a) Plot the points JC5, 8), K(3, 6) and L(7, 5) on a graph paper.
(b) Join each of the points to the origin.
(c) Draw line segments passing through all the points plotted.
(d) Find the position vectors of the points lying on the line segments.
(e) Explain the relationship between any two vectors in (d).

Showing collinearity of directed vectors Activity 4.6(b) (Work in groups)

e
ICT Activity
(a) In your environment, take photographs of different areas where vectors are applied.
(b) Transfer your photographs to a computer.
(c) Explain to the rest of the class, the process of transferring photographs.