Vectors and Angles

When you are solving a physics problem, especially a realistic one, you will be given angles. In the last section we have learned that vectors look like this: 574m/s [E66°N] where 66° is the angle going from the horizontal East direction towards the vertical North direction.

Most of the time you will have to simplify complicated physics situation into several simple triangles, representing the addition of several magnitudes and the sum.

In a setup such as this you will be able to use angles that are given to you not only to find a magnitude but also its direction. To find missing vector quantities you need to be proficient with trigonometry. Lets review some trigonometric formulae, where ß is an angle in a triangle:

Right triangles

sin ß = opposite / hypotenuse
cos ß = adjacent / hypotenuse
tan ß = opposite / adjacent

Other triangles

a² = b² × c² - 2ac cos( A );
a/sin A = b/sin B = c/sin C

Lets plunge into an example invloving vectors and angles

The Thompsons drive to their friends' lodge but they do not take a straight path. They travel at a velocity of 70km/h [W20°N] for and hour and then they stop and travel at a velocity of 80km/h [N20°E] for half an hour. Using the data given find the displacement of the lodge in relation to the Thompsons' house.

Since we are to find the displacement of the lodge ( how far is it, and what angle it is from the house when travelling there on a straight path ) we convert the velocities and times given into displacements.

1 = 70km/h [W20°N] × 1h
1 = 70km [W20°N]

2 = 80km/h [N20°E] × 1/2 h
2 = 40km [N20°E]

= 1 + 2

Our resultant displacement is the sum of the two other displacements. Lets sketch the situation so we can better understand the angle at which the lodge is to the Thompsons' house.

The black vectors are the two displacements through which the Thompsons went to get to the lodge, the blue vector is the shorter, direct path to the lodge ( the resultant vector ). When looking at the angles in relation to the directions ( the "cross" ) we can deduct that the angle between the two displacements is 90°.

This is convenient for us, as we can simply use the Pythagorean theorem to get the magnitude of , the direct distance to the lodge.

||² = |1|² + |2|²
||² = 70² + 40²
||² = 6500
|| = 80.622 km

The direct distance to the lodge is 81 km ( significant digits ). Since we have used the absolute value, | |, of the resultant displacement we don't have an angle yet ( we need the angle to know exactly in which direction to drive ). Since we know the magnitude of all three sides of our "triangle" we can now use a trigonometric ratio to find any angle. The angle we are interested in is the angle between the first displacement and the resultant displacement, when we add 20° to it we will be able to say that the direction is [West 20° + ?° N].

Let ß be the angle between 1 and
Cos ß = 1 /
Cos ß = 70 / 81
Cos ß = 0.8641
ß = 30°

The angle of the displacement is 20° + 30° = 50°
We can now state the displacement of the lodge from the Thompson's
house as: 81km [W50°N].
You can also state the direction as going from north to west, [N40°W].

Components

The above example was simple to solve because when we drew the situation, there was a 90° angle and we could use trigonometric ratios. But what about situations where the angles between the vectors are irregular? To greatly simplify addition of multiple vectors and operations with angles we use components.

Just like the name says, components are the simpler vectors of which each vector is made of. Each can be represented as the sum of two vectors, one in the up/down direction and one in the left/right direction. Look at the picture below to see how you can make a "complex" vector out of two "simple" ones.

The grid makes it easy to see that to make all of the blue vectors we just used vertical and horizontal vectorsd. The three vectors on the right show that to make a vector larger while keeping its direction the same we elongate both component vectors by the same amount.

Using Components

We have the vector 60m/s² [S22°E] ( acceleration in a certain direction ) how do we break it down into simpler components? Lets draw the situation, remember that the addition of two vectors at right angles to each other looks like a right triangle.

We know all the angles, but we'll focus on 22°, and we know the length of our hypotenuse ( 60m ) therefore we can discover the other two vectors ( "sides of the triangle" ) using trigonometry.

The south component has a magnitude of 60 × Cos 22°
which is 56m. The east component has a magnitude of 60 × Sin 22°
which is 22m.
Therefore:
60m [S22°E] = 60 × Cos 22° + 60 × Sin 22°
60m [S22°E] = 56m [S] + 22m [E]

Every vector can be represented as the addition of two other vectors

Adding Multiple Vectors

When we are adding a large amount of vectors ( a complex vector motion)
we can break down each vector into components and then add the components:

A radar outpost in Hawaii detects a UFO, it can turn on a dime and it's motion is sporadic. The outpost gathered a series of veclocities and times which you have to add together and find out the UFO's resultant displacement. The displacements began being recorded once the UFO was above the outpost, so the motion can be related to the outpost.

a) Find the resultant displacement, use components.
b) What does the resultant displacement tell us?
c) Does it matter in which order the velocities are added?
d) What is the average velocity? What does it mean?

Velocities:
320km/h [N76°E] for 30 min
340km/h [E35°S] for 20 min
100km/h [N40°W] for 15 min
20km/h [S30°W] for 1 hour

a)
Lets find the displacements, multiply the velocities by the amount of hours
during which the UFO travelled at those velocities. ( 15 min = 1/4 hours )

160km [N76°E]
113km [E35°S]
25km [N40°W]
20km [S30°W]

Break down the displacements into components and add: =
160 × Cos 76° [N] + 160 × Sin 76° [E] +
113 × Cos 35° [E] + 113 × Sin 35° [S] +
25 × Cos 40° [N] + 25 × Sin 40° [W] +
20 × Cos 30° [S] + 20 × Sin 30° [W]

=
38.7 [N] + 155.2 [E] + 92.6 [E] + 64.8 [S] + 19.2 [N] +
16.1 [W] + 17.3 [S] + 10.0 [W]

Remember that movement west is just a negative movement east, and that a movement north is a backwards movement south. We should get all horizontal motion in terms of one direction, and likewise for vertical motion. Lets write all vertical motion in terms of north and all horizontal motion in terms of east.

=
38.7 [N] + 155.2 [E] + 92.6 [E] - 64.8 [N] + 19.2 [N] -
16.1 [E] - 17.3 [N] - 10.0 [E]
= -24.2 [N] + 221.7[E]
= 24.2 [S] + 221.7[E]

Now that we have the two components of the resultant displacement we can use the pythagorean theorem and trigonometry to find the magnitude of and its direction.
Since the components form a 90° angle betweeen them:
||² = 24.2² + 221.7²
|| = 223 km

Let ß be the angle between and the east direction
Tan ß = 24.2 [S] / 221.7[E]
Tan ß = 24.2 / 221.7
Tan ß = 0.10915
ß = 6.229

Therefore the UFO's resultant displacement is 223km [E6°S] from the radar outpost. We first calculated how far is it, and then we calculated the angle of the resultant displacement away from its east component and toward its south component.

b) What does the displacement tell us?
Doing all of this work for nothing is not very rewarding, you should always know what you are doing and why you are doing it.
We have just added a series of motions together, they naturally occured head-to-tail, sequentially. The resultant displacement is the final spot at which the UFO was sighted, we can now dispatch a research team to that location. Remember that the resultant displacement is relative to the radar outpost!
Also, if the velocities began being recorded before/after the UFO was above the outpost we would not know to what initial position, on the ground, we should relate displacements to. This may sound confusing at first but by understanding reference points this will become clear.

c) Does the order of addition matter?
No. When adding vectors the order of addition does not matter. Vectors are concerned with "where you end up" as opposed to "how you got there" and every arrangement of the same set of motions will finish in the same place.

d) What is the average velocity? What does it mean?
Velocity is the change in displacement over time. The change in position was 223km [E6°S], the total time was
30min + 20min + 15min + 60min = 125min = 2.08hours.
Average velocity = 223km [E6°S] / 2.083hours
Average velocity = 107km/h [E6°S]

The average velocity means that if the UFO went in a straight line from the outpost to its final position, it would have to travel at 107km/h [E6°S] to arrive there after the same time it took for the sporadic motion. This is not very useful for this question but it might be for other types.

Conclusion

For all questions but the simplest you should create a vector diagram of the situation. In most cases your diagram will be a triangle. Use your knowledge of triangles, the pythagorean theorem, trigonometric ratios and the sine and cosine laws to deduct magnitudes and directions of vectors.

When faced with vectors that are not unidirectional ( N and S, E and W ) you should use components and the pythagorean theorem to find magnitudes. Don't forget to find out the direction angle of each vector as well as its magnitude.

When adding component vectors you should write all vectors in terms of two directions, all vertical in north or south, all horizontal in east or west. Motion in the other direction should be considered as negative motion in your chosen direction.

Vectors are very useful for many different types of problems and they will be used extensively in the following sections, the next one of which is Uniform Acceleration.