Maths

Vectors - Higher

A vector describes a movement from one point to another.

# Vector notation

A vector quantity has both direction and magnitude (size).

(In contrast a scalar quantity has magnitude only - eg, the numbers 1, 2, 3, 4...)

For example this arrow represents a vector. The direction is given by the arrow, while the length of the line represents the magnitude.

This vector can be written as: , a, or .

In print, a is written in bold type. In handwriting, the vector is indicated by putting a squiggle underneath the letter:

Question

Write down the 3 ways to describe the vector if the arrow is now pointing from B to A.

Answer

Remember that the arrow describes the direction. So, in this case, the vector is from B to A. If we move 'backwards' along a vector, it becomes negative, so a becomes -a. Moving from B to A entails moving 3 units to the left, and 4 down.

So the three ways to write this vector are:

, -a and

# Vector 'arithmetic'

## Equal vectors

If two vectors have the same magnitude and direction, then they are equal.

## Adding vectors

Look at the graph below to see the movements between PQ, QR and PR.

Vector followed by vector represents a movement from P to R.

Written out the vector addition looks like this

## Subtracting vectors

Subtracting a vector is the same as adding a negative version of the vector (remember that making a vector negative means reversing its direction).

Look at the diagram and imagine going from X to Z. How would you write the path in vectors using only the vectors and ?

You could say it is vector followed by a backwards movement along .

So we can write the path from X to Z as

Written out in numbers it looks like this:

Question

If x = , y = and z = , find:

1. -y
2. x - y
3. 2x + 3z

Answer
1. (Did you remember to change the signs?)

## Resultant vectors

To travel from X to Z, it is possible to move along vector followed by . It is also possible to go directly along .

is therefore known as the resultant of and .

# Geometric problems

Question

Write as single vectors:

1. f + g
2. a + b
3. e - b - a

Answer
1. e
2. -c (Did you remember the minus sign?)
3. -d

Remember

Two vectors are equal if they have the same magnitude and direction, regardless of where they are on the page. You need to use this fact to answer the next question.

Question

Triangles ABC and XYZ are equilateral.

X is the midpoint of AB, Y is the midpoint of BC, Z is the midpoint of AC.

= a , = b , = c

Express each of the following in terms of a, b and c.

Answer
1. c
2. -aRemember that is parallel to and of the same length, but the direction is different.

3. b + c

(It is also possible to move from X to A, and then on to C. This would give the answer -a + 2c. How many other answers can you think of?)

4. b - a

Or 2b - c,

or -2a + c

5. 2c

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