Motion and Vectors in Two Dimensions

Vectors in One and Two Dimensions

We have previously seen that many quantities are described as vectors – that is they have both magnitude and direction. For example: displacement, velocity and acceleration. Many of the examples studied so far have considered vectors in one-dimension and directions have been described as up or down, left or right, east or west etc.

Motion is often not restricted to moving in one direction and this may require the addition or subtraction of vectors in two different directions to determine some quantity or value. This section considers vectors that may act be in two dimensions. This requires an analysis of individual vectors and their components in order to manipulate and work with multiple vector problems.

Resolving Vectors into Perpendicular Components

A vector that lies in a two dimensional plane can be broken down into its components. Common practice is to break the vector into perpendicular components. Depending on the situation, these perpendicular components may be described as compass bearings (north, south, east or west) if we are analysing a car driving along the road. We may be considering a ball thrown through the air and describe its components as vertical and horizontal.

However a problem is analysed, the two vector components should always be perpendicular. These can be drawn on your page when analysing problems and then trigonometry and Pythagoras’ theorem can be applied to resolving the components:

Consider the vector \overset { \rightarrow }{ c } below:

 \overset { \rightarrow }{ c }  can be expressed as the sum of two vectors, \overset { \rightarrow }{ { c }_{ x } } and \overset { \rightarrow }{ { c }_{ y } } perpendicular to each other, one in the horizontal direction and the other in the vertical.

\overset { \rightarrow }{ c } =\overset { \rightarrow }{ { c }_{ x } } +\overset { \rightarrow }{ { c }_{ y } }

The magnitude of these vectors can be determined by using trigonometry:

 { c }_{ x }=\quad c\cos { \theta }

{ c }_{ y }=\quad c\sin { \theta }

It is important to remember that direction must also be considered when resolving vectors into their components, especially when analysing and solving problems involving two separate vectors. Directions in many vector problems are given as compass bearings so you need to be confident with determining directions and angles given a compass bearing.

Adding Vector Components

The opposite of resolving a vector into its components is the addition of components to obtain a vector. Assuming that vector components are perpendicular to each other, Pythagoras’s theorem can be applied to solve for the vector. Trigonometry also needs to be applied to find any angle and an appropriate direction given as a compass bearing.

c=\sqrt { { { c }_{ x } }^{ 2 }+{ { c }_{ y } }^{ 2 } }

\sin { \theta } =\cfrac { opp }{ hyp }

\cos { \theta } =\cfrac { adj }{ hyp }

\tan { \theta } =\cfrac { opp }{ adj }

Distance and Displacement on a Plane

Distance and displacement are two important and useful quantities that are used to describe motion. Recall that distance is a scalar quantity and displacement is a vector.

In the previous section it was seen that a single vector can be resolved into vector components. These vector components are themselves vectors that are just arranged in a convenient manner to allow for easier analysis of problems. A logical extension of this is that we can take two individual vectors and add them together to get a third vector. This is useful providing the vectors are the same type. Any number of vectors can be added together to calculate a single resultant vector.

The total displacement of an object can be determined by adding together all individual components of distance or displacement. For example, an object may move along a path and change direction several times. The total displacement can only be determined by analysing each vector and its components. The following diagram illustrates 3 different vectors:

These vectors can be analysed by looking at the vector components of each individual vector (blue):

The individual x and y components can be added to find the total sum of all three vectors (orange):

When adding vectors together they are placed head to tail from the first to the last vector. If analysing graphically, the vectors need to be drawn to scale and the angles measured correctly. Vector addition can also be analysed by resolving each vector into vertical and horizontal components and adding up the total parts of the x (horizontal) and y (vertical) sections. 

Describing Motion Using Vectors

Another useful application of vectors is analysing the motion or change in motion of an object, particularly velocity. We may need to combine the velocity of two objects relative to the ground in order to find the total velocity, or, we may need to find the change in velocity of an object as it changes direction and/or speed. The calculation of these two examples is done differently:

  • If a bus is driving along a road and a passenger walks to the front of the bus, we would need to add these vector components to determine the velocity of the passenger relative to the road:

{ v }={ v }_{ b }+{ v }_{ p }, where  v is the velocity of the passenger relative to the ground, { v }_{ b } is the velocity of the bus relative to the ground and { v }_{ p } is the velocity of the passenger relative to the bus.

  • If a sailing boat is moving at a certain velocity west and changes course to head north-east, we would need to subtract these vector components to determine the change is speed of the sailing boat relative to the water:

{ \Delta v }={ v }_{ f }-{ v }_{ i }, where  { \Delta v } is the change in velocity of the sailing boat, { v }_{ f } is the final velocity of the sailing boat and { v }_{ i } is the initial velocity of the sailing boat.

Example 1:

Determine the vertical and horizontal components of the velocity illustrated below:


{ v }_{ x }=15\cos { 34 }

12.4 m/s

{ v }_{ y }=15\sin { 34 }

8.4 m/s

Example 2:

A passenger on a train travelling at 10 m/s walks toward the front of the train at a speed of 1.5 m/s relative to the train. What is the velocity of the passenger relative to the ground?

{ v }={ v }_{ t }+{ v }_{ p }

{ v }=10+1.5

11.5 m/s