There are many ways of writing the symbol for a vector. Vectors are denoted by symbols with an arrow pointing to the right above it. For example, a→a→, v→v→ and F→F→ represent the vectors acceleration, velocity and force, meaning they have both a magnitude and a direction.

Sometimes just the magnitude of a vector is needed. In this case, the arrow is omitted. In other words, FF denotes the magnitude of the vector F→F→. |F→||F→| is another way of representing the magnitude of a vector.

Vectors are drawn as arrows. An arrow has both a magnitude (how long it is) and a direction (the direction in which it points). The starting point of a vector is known as the tail and the end point is known as the head.

Figure 1: Examples of vectors

Figure 2: Parts of a vector

The simplest method of expressing direction is with relative directions: to the left, to the right, forward, backward, up and down.

Another common method of expressing directions is to use the points of a compass: North, South, East, and West. If a vector does not point exactly in one of the compass directions, then we use an angle. For example, we can have a vector pointing 40∘∘ North of West. Start with the vector pointing along the West direction: Then rotate the vector towards the north until there is a 40∘∘ angle between the vector and the West. The direction of this vector can also be described as: W 40∘∘ N (West 40∘∘ North); or N 50∘∘ W (North 50∘∘ West)

Figure 3

Figure 4

Figure 5

The final method of expressing direction is to use a bearing. A bearing is a direction relative to a fixed point.

Given just an angle, the convention is to define the angle with respect to the North. So, a vector with a direction of 110∘∘ has been rotated clockwise 110∘∘ relative to the North. A bearing is always written as a three digit number, for example 275∘∘ or 080∘∘ (for 80∘∘).

Figure 6

Draw each of the following vectors to scale. Indicate the scale that you have used:

When vectors are added, we need to add both a magnitude and a direction. For example, take 2 steps in the forward direction, stop and then take another 3 steps in the forward direction. The first 2 steps is a displacement vector and the second 3 steps is also a displacement vector. If we did not stop after the first 2 steps, we would have taken 5 steps in the forward direction in total. Therefore, if we add the displacement vectors for 2 steps and 3 steps, we should get a total of 5 steps in the forward direction. Graphically, this can be seen by first following the first vector two steps forward and then following the second one three steps forward (ie. in the same direction):

Figure 13

We add the second vector at the end of the first vector, since this is where we now are after the first vector has acted. The vector from the tail of the first vector (the starting point) to the head of the last (the end point) is then the sum of the vectors. This is the head-to-tail method of vector addition.

As you can convince yourself, the order in which you add vectors does not matter. In the example above, if you decided to first go 3 steps forward and then another 2 steps forward, the end result would still be 5 steps forward.

The final answer when adding vectors is called the resultant. The resultant displacement in this case will be 5 steps forward.

Definition 3: Resultant of Vectors

The resultant of a number of vectors is the single vector whose effect is the same as the individual vectors acting together.

In other words, the individual vectors can be replaced by the resultant – the overall effect is the same. If vectors a→a→ and b→b→ have a resultant R→R→, this can be represented mathematically as,

Let us consider some more examples of vector addition using displacements. The arrows tell you how far to move and in what direction. Arrows to the right correspond to steps forward, while arrows to the left correspond to steps backward. Look at all of the examples below and check them.

Figure 14

This example says 1 step forward and then another step forward is the same as an arrow twice as long – two steps forward.

Figure 15

This examples says 1 step backward and then another step backward is the same as an arrow twice as long – two steps backward.

It is sometimes possible that you end up back where you started. In this case the net result of what you have done is that you have gone nowhere (your start and end points are at the same place). In this case, your resultant displacement is a vector with length zero units. We use the symbol 0→0→ to denote such a vector:

Figure 16

Figure 17

Check the following examples in the same way. Arrows up the page can be seen as steps left and arrows down the page as steps right.

Try a couple to convince yourself!

Table 1

Figure 18

Figure 19

Table 2

Figure 20

Figure 21

It is important to realise that the directions are not special– `forward and backwards' or `left and right' are treated in the same way. The same is true of any set of parallel directions:

Table 3

Figure 22

Figure 23

Table 4

Figure 24

Figure 25

In the above examples the separate displacements were parallel to one another. However the same head-to-tail technique of vector addition can be applied to vectors in any direction.

Table 5

Figure 26

Figure 27

Figure 28

Now you have discovered one use for vectors; describing resultant displacement – how far and in what direction you have travelled after a series of movements.

Although vector addition here has been demonstrated with displacements, all vectors behave in exactly the same way. Thus, if given a number of forces acting on a body you can use the same method to determine the resultant force acting on the body. We will return to vector addition in more detail later.

What does it mean to subtract a vector? Well this is really simple; if we have 5 apples and we subtract 3 apples, we have only 2 apples left. Now lets work in steps; if we take 5 steps forward and then subtract 3 steps forward we are left with only two steps forward:

Figure 29

What have we done? You originally took 5 steps forward but then you took 3 steps back. That backward displacement would be represented by an arrow pointing to the left (backwards) with length 3. The net result of adding these two vectors is 2 steps forward:

Figure 30

Thus, subtracting a vector from another is the same as adding a vector in the opposite direction (i.e. subtracting 3 steps forwards is the same as adding 3 steps backwards).

In the problem, motion in the forward direction has been represented by an arrow to the right. Arrows to the right are positive and arrows to the left are negative. More generally, vectors in opposite directions differ in sign (i.e. if we define up as positive, then vectors acting down are negative). Thus, changing the sign of a vector simply reverses its direction:

Table 6

Figure 31

Figure 32

Table 7

Figure 33

Figure 34

Table 8

Figure 35

Figure 36

In mathematical form, subtracting a→a→ from b→b→ gives a new vector c→c→:

This clearly shows that subtracting vector a→a→ from b→b→ is the same as adding (-a→)(-a→) to b→b→. Look at the following examples of vector subtraction.

Figure 37

Figure 38

What happens when you multiply a vector by a scalar (an ordinary number)?

Going back to normal multiplication we know that 2×22×2 is just 2 groups of 2 added together to give 4. We can adopt a similar approach to understand how vector multiplication works.

Figure 39

Graphical techniques involve drawing accurate scale diagrams to denote individual vectors and their resultants. We next discuss the two primary graphical techniques, the head-to-tail technique and the parallelogram method.

The Parallelogram Method

The parallelogram method is another graphical technique of finding the resultant of two vectors.

Method: The Parallelogram Method

1. Make a rough sketch of the vector diagram.
2. Choose a scale and a reference direction.
3. Choose either of the vectors to be added and draw it as an arrow of the correct length in the correct direction.
4. Draw the second vector as an arrow of the correct length in the correct direction from the tail of the first vector.
5. Complete the parallelogram formed by these two vectors.
6. The resultant is then the diagonal of the parallelogram. The magnitude can be determined from the length of its arrow using the scale. The direction too can be determined from the scale diagram.

Exercise 4: Parallelogram Method of Vector Addition I

A force of F1=5NF1=5N is applied to a block in a horizontal direction. A second force F2=4NF2=4N is applied to the object at an angle of 30∘∘ above the horizontal.

Figure 50

Determine the resultant force acting on the block using the parallelogram method of accurate construction.

Solution

1. Step 1. Firstly make a rough sketch of the vector diagram :

   Figure 51

2. Step 2. Choose a suitable scale :

   In this problem a scale of 1 cm = 1 N would be appropriate, since then the vector diagram would take up a reasonable fraction of the page. We can now begin the accurate scale diagram.

3. Step 3. Draw the first scaled vector :

   Let us draw F1F1 first. According to the scale it has length 5 cm.

   Figure 52

4. Step 4. Draw the second scaled vector :

   Next we draw F2F2. According to the scale it has length 4 cm. We make use of a protractor to draw this vector at 30∘∘ to the horizontal.

   Figure 53

5. Step 5. Determine the resultant vector :

   Next we complete the parallelogram and draw the diagonal.

   Figure 54

   The resultant has a measured length of 8,7 cm.

6. Step 6. Find the direction :

   We use a protractor to measure the angle between the horizontal and the resultant. We get 13,3∘∘.

7. Step 7. Apply the scale conversion :

   Finally we use the scale to convert the measured length into the actual magnitude. Since 1 cm = 1 N, 8,7 cm represents 8,7 N. Therefore the resultant force is 8,7 N at 13,3∘∘ above the horizontal.

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The parallelogram method is restricted to the addition of just two vectors. However, it is arguably the most intuitive way of adding two forces acting on a point.

As a further example of components let us consider a block of mass mm placed on a frictionless surface inclined at some angle θθ to the horizontal. The block will obviously slide down the incline, but what causes this motion?

The forces acting on the block are its weight mgmg and the normal force NN exerted by the surface on the object. These two forces are shown in the diagram below.

Figure 63

Now the object's weight can be resolved into components parallel and perpendicular to the inclined surface. These components are shown as dashed arrows in the diagram above and are at right angles to each other. The components have been drawn acting from the same point. Applying the parallelogram method, the two components of the block's weight sum to the weight vector.

To find the components in terms of the weight we can use trigonometry:

The component of the weight perpendicular to the slope Fg⊥Fg⊥ exactly balances the normal force NN exerted by the surface. The parallel component, however, Fg∥Fg∥ is unbalanced and causes the block to slide down the slope.

Components can also be used to find the resultant of vectors. This technique can be applied to both graphical and algebraic methods of finding the resultant. The method is simple: make a rough sketch of the problem, find the horizontal and vertical components of each vector, find the sum of all horizontal components and the sum of all the vertical components and then use them to find the resultant.

Consider the two vectors, A→A→ and B→B→, in Figure 65, together with their resultant, R→R→.

Figure 65: An example of two vectors being added to give a resultant

Each vector in Figure 65 can be broken down into one component in the xx-direction (horizontal) and one in the yy-direction (vertical). These components are two vectors which when added give you the original vector as the resultant. This is shown in Figure 66 where we can see that:

In summary, addition of the xx components of the two original vectors gives the xx component of the resultant. The same applies to the yy components. So if we just added all the components together we would get the same answer! This is another important property of vectors.

Figure 66: Adding vectors using components.

Vector Multiplication

Vectors are special, they are more than just numbers. This means that multiplying vectors is not necessarily the same as just multiplying their magnitudes. There are two different types of multiplication defined for vectors. You can find the dot product of two vectors or the cross product.

The dot product is most similar to regular multiplication between scalars. To take the dot product of two vectors, you just multiply their magnitudes to get out a scalar answer. The mathematical definition of the dot product is:

a → • b → = | a → | · | b → | cos θ a → • b → = | a → | · | b → | cos θ

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Take two vectors a→a→ and b→b→:

Figure 70

You can draw in the component of b→b→ that is parallel to a→a→:

Figure 71

In this way we can arrive at the definition of the dot product. You find how much of b→b→ is lined up with a→a→ by finding the component of b→b→ parallel to a→a→. Then multiply the magnitude of that component, |b→||b→|cosθcosθ, with the magnitude of a→a→ to get a scalar.

The second type of multiplication, the cross product, is more subtle and uses the directions of the vectors in a more complicated way. The cross product of two vectors, a→a→ and b→b→, is written a→×b→a→×b→ and the result of this operation on a→a→ and b→b→ is another vector. The magnitude of the cross product of these two vectors is:

| a → × b → | = | a → | | b → | sin θ | a → × b → | = | a → | | b → | sin θ

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We still need to find the direction of a→×b→a→×b→. We do this by applying the right hand rule.

Method: Right Hand Rule

1. Using your right hand:
2. Point your index finger in the direction of a→a→.
3. Point the middle finger in the direction of b→b→.
4. Your thumb will show the direction of a→×b→a→×b→.

Figure 72