Basic Vectors

Basic Vectors
What are vectors?
 A vector is a type of number that has both a size and a direction

 Here we only deal with two-dimensional vectors, although it is possible to have vectors
with any number of dimensions
Representing vectors
 Vectors are represented as arrows, with the arrowhead indicating the direction of the
vector, and the length of the arrow indicating the vector’s magnitude (ie its size):
 In print vectors are usually represented by bold letters (as with vector a in the diagram
above), although in handwritten workings underlined letters are normally used, a.
 Another way to indicate a vector is to write its starting and ending points with an arrow
symbol over the top such as

 Note that the order of the letters is important! Vector in the above diagram would
point in the opposite direction (ie with its ‘tail’ at point B, and the arrowhead at point A).
Vectors and transformation geometry
In transformation geometry, translations are indicated in the form of a column vector:

 In the following diagram, Shape A has been translated six squares to the right and 3
squares up to create Shape B
 This transformation is indicated by the translation vector :
 Note: ‘Vector’ is a word from Latin that means ‘carrier’

 In this case, the vector ‘carries’ shape A to shape B, so that meaning makes perfect sense!
Vectors on a grid
 You also need to be able to work with vectors on their own, outside of the transformation
geometry context
 When vectors are drawn on a grid (with or without x and y axes), the vectors can be
represented in the same (x y) column vector form as above
Multiplying a vector by a scalar
 A scalar is a number with a magnitude but no direction – ie the regular numbers you are
used to using
 When a vector is multiplied by a positive scalar, the magnitude of the vector changes, but
its direction stays the same
 If the vector is represented as a column vector, then each of the numbers in the column
vector gets multiplied by the scalar
 Note that multiplying by a negative scalar also changes the direction of the vector:
 Note in particular that vector -a is the the same size as vector a, but points in the opposite
direction!
Adding and subtracting vectors
 Adding two vectors is defined geometrically, like this:

 Subtracting one vector from another is thought of as adding a negative vector
a – b = a + (-b)
 When vectors are represented as column vectors, adding or subtracting is simply a

matter of adding or subtracting the vectors’ x and y coordinates
 For example:
Worked example
The points A, B and C are shown on the following coordinate grid.
(a)
Write the vectors and as column vectors.
Start by drawing the three vectors onto the grid.

From A to B, it is 6 to the right and 2 up.
From A to C, it is 7 to the right and 6 down.
From C to B, it is 1 to the left and 8 up.

(b)
Using the column vectors from (a), confirm that .
Just perform the subtraction on the column vectors.
Did this video help you?

YesNo
Length of a Vector
What is a vector?
 Vectors have various uses in mathematics

o In mechanics vectors represent velocity, acceleration and forces
o At GCSE vectors are used in geometry – eg. translation
o Ensure you are familiar with the Revision Notes Vectors – Basics
 These notes look at finding the length (also referred to as the magnitude, or modulus),
of a vector
o Vectors are given in column vector form
o Vectors have length (magnitude) and direction
What is the length of a vector?
 This depends on the use of the vector

o For velocity, magnitude would be speed
o For a force, magnitude would be the strength of the force (in Newtons)
 The words length, magnitude and modulus mean the same thing with vectors
 In geometry magnitude and modulus mean the length or distance of the vector
o This is always a positive value
o The direction of the vector is irrelevant
 Magnitude or modulus is indicated by vertical lines
o |a| would mean the magnitude of vector a
o You do not need to be use or understand this notation for your exam
How do I find the length of a vector

 Pythagoras’ Theorem!
Exam Tip
 Sketch a vector to help, it does not have to be to scale, then you can use this to form a
right-angled triangle.
Worked example
Two points, P and Q, are plotted on a grid. Given that find the length of the line segment
that joins P and Q.
You can form a right angled triangle by starting at P and then going 8 to the right and 6 down to
get to Q.
The length between P and Q is the hypotenuse of this triangle so you can use Pythagoras'
theorem.
Length is 10 units

Basic Vectors

Uploaded by

Copyright:

Available Formats

Basic Vectors

Uploaded by

Document Information

Original Description:

Copyright

Available Formats

Share this document

Share or Embed Document

Sharing Options

Did you find this document useful?

Is this content inappropriate?

Copyright:

Available Formats

Basic Vectors

Uploaded by

Copyright:

Available Formats

Basic Vectors

What are vectors?

 A vector is a type of number that has both a size and a direction

symbol over the top such as

Vectors and transformation geometry

In transformation geometry, translations are indicated in the form of a column vector:

 This transformation is indicated by the translation vector :

 Note: ‘Vector’ is a word from Latin that means ‘carrier’

Adding and subtracting vectors

 Adding two vectors is defined geometrically, like this:

 When vectors are represented as column vectors, adding or subtracting is simply a

Write the vectors and as column vectors.

Start by drawing the three vectors onto the grid.

From A to C, it is 7 to the right and 6 down.

From C to B, it is 1 to the left and 8 up.

Using the column vectors from (a), confirm that .

Just perform the subtraction on the column vectors.

Did this video help you?

 Vectors have various uses in mathematics

What is the length of a vector?

 This depends on the use of the vector

How do I find the length of a vector

You might also like