What Is Meant By Drawing A Vector To Scale

Vectors and scalars

All physical quantities fall broadly into two categories, scalars and vectors. Scalars are quantities that have size or magnitude only, but no given direction. For example, the mass of an object can be fully described as 45 kg, and no further information is needed. Vectors have both magnitude and direction.

The direction is usually given as either + (positive) and - (negative) or as an angle to the horizontal or vertical. Other directions can be used, such as points on a compass, up, down backwards forwards etc.

When describing a vector both its magnitude and direction must be stated, and they can be drawn using arrows to show the direction.

Here is a list of common vectors and scalars that you will use during your A-level studies.

Scalar	Vector
speed	velocity
mass	force
energy	momentum
distance	displacement
pressure	moment
frequency	field strength
time
volume
charge
current
temperature
power

As you can see, many scalar quantities have scalar equivalents, such as speed and velocity. One of the simplest examples of the difference between scalars and vectors is in the case on distance and displacement. This idea might be familiar to students from GCSE maths. Whereas distance covered by an object when travelling is the length of its path the displacement is the length of the shortest path from the origin of the journey to the end. In a circular path of motion whereas the distance would be π times the diameter of the circle travelled the displacement would be 0. This is because the object starts and ends at the same point in space.

the difference between distance and displacement — **Figure 1:** The difference between distance and displacement

When two or more vectors act on an object the overall effect they have on the object is called the resultant and can itself be described as a vector.

Using vectors leads to some different arithmetic; we can no longer always just add vector quantities together as we are used to. We must take into account their direction.

When the vectors are acting in one dimension (along the same plane) we can use + and - as the directions and then addition of vectors is not too difficult.

adding two parallel vectors — **Figure 2:** Adding two parallel vectors.

In the top diagram both of the force vectors are acting in the same direction so we can simply add them together. In the bottom diagram the two force vectors are acting in opposite directions, som we must consider their relative directions. (It doesn't matter which direction we call positive or which direction we call negative, the result will still be the same.) In this example the force acting to the right (7 N) is positive and the force acting to the left (5 N) is negative. When we have assigned the signs we can then add them together. It is important that you make sure that the directional signs have been assigned and is consistent throughout you working as the answer's sign tells you the direction of the final or resultant vector. In this case the resultant vector is +2N which tells us that the force is acting to the right.

When vectors are not acting along a plane we have one of two options:

Use trigonometry
Draw a scale diagram

There are different situations where each approach is more appropriate, and we will look at each in turn.

Using trigonometry to resolve vectors

When two vectors are acting at right angles to each other we need to use pythagoras' theorem to find the resultant. The two vectors make up two sides of a right angled triangle so it is a simple matter to find the third side of the triangle.

adding two vectors at right angles — **Figure 3:** Calculating the resultant of two vectors at right angles to each other.

If two vectors acting at right angles to each other can produce a single resultant vector, then it should be clear that the reverse can also be true. Any individual vector can be resolved into two components at right angles to each other.

To do this we need to know the angle at which the vector is acting to some reference point. Once we know the angle we can then trigonometry to resolve the vector into its components.

resolving a vectot into right angled componets — **Figure 4:** Any vector quantity can be broken into two compoents which are at right angles to each other.

In this example a ball has been launched with a velocity (s)of 40 ms^-1 at an angle of 30 degrees to the horizontal. This velocity vector can be resolved into vertical and horizontal components.

The horizontal component (H) can be represented as the adjacent side of a right angled triangle and the vertical component (V) is the opposite side of the triangle.

The horizontal component is therefore:

$$\large\textrm{H}=40 \cos (30)=34.6\textrm{ms}^{-1}$$

And the vertical component is:

$$\large\textrm{H}=40 \sin (30)=20.0 \textrm{ms}^{-1}$$

This is very useful when the two components of a vector need to be considered separately, as is often the case with forces and projectile motion.

Using a scale diagram to solve vectors

In situations when two vectors are not acting at right angles is may be necessary to draw a scale to find the resultant. Scale diagrams must be drawn using a sharp pencil, ruler and an angle measurer. (Make sure that you take these items into your exam!)

In this example two forces are acting at a point. One 16N force and one 12N force.
A suitable scale of 1cm force 2N is chosen.
As the two forces are both acting at a point the resultant force can be found by drawing a parallelogram of forces.

drawing a scale diagram part 1 — **Figure 5:** Drawing a scale diagram, step 1. Carefully draw the vectors.

The angle between the two forces is carefully measured and the two other sides of the parallelogram can be drawn

drawing a scale diagram part 2 — **Figure 6:** Drawing a scale diagram, step 2. Complete the parallelogram

Once the parallelogram has been drawn the resultant force can be drawn as the diagonal starting from the origin of the two forces.
Its magnitude can be measured directly using a ruler and converted using the scale, and it's direction found using protractor.

drawing a scale diagram part 3 — **Figure 7:** Drawing a scale diagram, step 3. Draw and measure the diagonal.

Scale diagrams need to be drawn carefully, and you need to choose as large a scale as possible to reduce any error.

(Many of the situations that require you to draw a scale diagram could be solved using the sine rule, but you will be expected to draw a diagram, and cannot receive full marks unless you do so!)