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By Tinashe Chikoko and Samson Mujakwi
It is back to school once again and this term is a serious one. Books should be your best friend. In this 21st century many teenagers are much into technology.
Did you know you can study your mathematics or favourite subject on compact disc or DVD?
This phenomenon which some are familiar with has seen most students buying discs in the streets of their favourite movies or download them on the internet.
It is fine but have you considered taking the initiative to academic?
We as the new team in town called Good News Electronic Academic Solutions are in a process to empower and educate the youths through DVDs.
We believe that education in Zimbabwe must be fully digitalised as our academic solutions complement convectional teaching methods by the use of academic DVDs. For a start we are concerned with the decline in the national mathematics pass rate and the shortage of mathematics and science teachers hence our DVDs can help in alleviating these challenges.
Interesting to note, our product has been inspected and endorsed by renowned academics from the University of Zimbabwe.
A lot of students are easily shifting from the text book reading to DVDs which helps because most of them love the television.
This week we pre-empt the Mathematics – Vectors, on our DVD.
Mathematics is fun.
Vectors
The term vector refers to quantities with magnitude and direction.
The opposite of a vector quantity is called a scalar.
Magnitude refers to size (how long the vector is) and examples of vector quantities include velocity, acceleration, weight, displacement among others.
A vector can be represented by a line segment. There are some basic mathematical principles that you need to understand prior to attempting the topic which include the following among others.
a) A strong and solid primary and ZJC mathematics background.
b) To know the concepts of ratio, proportion and scale.
c) To solve triangles using the Pythagoras theorem.
d) To solve simultaneous linear equations using both the substitution and elimination methods.
Many students are often reluctant to tackle questions using vectors.
We think this is partly because often vectors is not taught until quite a way through a school maths course, so they are unfamiliar.
This short article aims to highlight some of the powerful techniques that can be used to solve problems involving vectors, and to encourage you to have a go at such problems to become more familiar with vector properties and applications.
So what are vectors?
When we first meet them, it’s often in the context of transformations – a translation can be expressed as a vector telling us how far something is translated to the right (or left) and up (or down).
Confusion can strike when we come across vectors being used to indicate absolute position relative to an origin as well as showing a direction.
Then we may be informed that a vector is ‘simply’ a quantity that has both magnitude and direction (unlike a scalar which only has magnitude).
Diagrams
It is helpful to separate out some of these ideas about vectors in order to make sense of things. Diagrams make it much easier to make sense of what is going on – one can represent a position vector as a point on the diagram with a line segment coming from the origin.
Direction vectors just become line segments joined onto other vectors, with a helpful arrow to remind me that a and ?a are in opposite directions!
Sometimes it’s useful to draw on lines parallel and perpendicular to my coordinate axes so can make sense of the x and y components of a vector.
Given a vector problem, a quick sketch can help you to see what’s going on, and the act of transferring the problem from the written word to a diagram can often give you some insight that will help you to find a solution.
Start by solving vector problems in two dimensions – it’s easier to draw the diagrams – and then move on to three dimensions.
(For four or more dimensions, it becomes more difficult to visualise!)
What to do when you get stuck. Here is a brief checklist of ideas to think about if you are stuck on a vector question, and drawing a diagram hasn’t helped.
Parallel vectors
Do you know about any parallel lines? Vector questions can often be about geometrical shapes like trapezia, rhombuses or parallelograms.
If two vectors are parallel, it can be really useful to express one in terms of the other – if a and b are parallel, try writing b=ka for some constant k.
Scalar products
Scalar products are immensely useful!
Sometimes if you’re at a loss to know what to do with vectors and vector equations, it’s worth just taking the scalar product of the whole equation with one of your vectors and seeing what you end up with.
Remember, a.a=|a|2, and if two vectors are perpendicular, their scalar product is 0.
Magnitude and direction
Some vector problems involve a vector function which tells you how an object’s position changes in time, for example. Working out how the magnitude and direction change over time can help you to picture the situation.
Vector equation of a line
Some students are intimidated by the vector equation of a line when they first meet it. We are very used to expressing lines using cartesian geometry in the form y=mx+c and other variants.
The vector equation of a line is no more complicated really, it’s just a case of getting used to it. In simple terms, lines are represented using vectors by specifying a point on the line with a position vector, and then using a direction vector to specify the direction of the line. In the same way that y=mx+c specifies a line that passes through (0,c) and has gradient m, the vector equation r=a+?b specifies a line that passes through the point with position vector a in the direction of b.
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