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Resultant of perpendicular vectors

In grade 10 you learnt about the resultant vector in one dimension, we are going to extend this to two dimensions. As a reminder, if you have a number of vectors (think forces for now) acting at the same time you can represent the result of all of them together with a single vector known as the resultant. The resultant vector will have the same effect as all the vectors adding together.

We will focus on examples involving forces but it is very important to remember that this applies to all physical quantities that can be described by vectors, forces, displacements, accelerations, velocities and more.

Vectors on the Cartesian plane

The first thing to make a note of is that in Grade 10 we worked with vectors all acting in a line, on a single axis. We are now going to go further and start to deal with two dimensions. We can represent this by using the Cartesian plane which consists of two perpendicular (at a right angle) axes. The axes are a -axis and a -axis. We normally draw the -axis from left to right (horizontally) and the -axis up and down (vertically).

We can draw vectors on the Cartesian plane. For example, if we have a force, , of magnitude 2 N acting in the positive -direction we can draw it as a vector on the Cartesian plane.

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Notice that the length of the vector as measured using the axes is 2, the magnitude specified. A vector doesn't have to start at the origin but can be placed anywhere on the Cartesian plane. Where a vector starts on the plane doesn't affect the physical quantity as long as the magnitude and direction remain the same. That means that all of the vectors in the diagram below can represent the same force. This property is know as equality of vectors.

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In the diagram the vectors have the same magnitude because the arrows are the same length and they have the same direction. They are all parallel to the -direction and parallel to each other.

This applies equally in the -direction. For example, if we have a force, , of magnitude 2,5 N acting in the positive -direction we can draw it as a vector on the Cartesian plane.

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Just as in the case of the -direction, a vector doesn't have to start at the origin but can be placed anywhere on the Cartesian plane. All of the vectors in the diagram below can represent the same force.

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The following diagram shows an example of four force vectors, two vectors that are parallel to each other and the -axis as well as two that are parallel to each other and the -axis.

To emphasise that the vectors are perpendicular you can see in the figure below that when originating from the same point the vector are at right angles.

Exercise 1:

Draw the following forces as vectors on the Cartesian plane originating at the origin:

  • in the positive -direction
  • in the positive -direction

We choose a scale of 1 N:1 cm. Now we sketch the vectors on the Cartesian plane (we can place the vectors anywhere on the Cartesian plane, we will place them starting at the origin):

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Draw the following forces as vectors on the Cartesian plane:

  • in the positive -direction
  • in the negative -direction
  • in the positive -direction

We choose a scale of 1 N:1 cm. Now we sketch the vectors on the Cartesian plane (we can place the vectors anywhere on the Cartesian plane):

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Draw the following forces as vectors on the Cartesian plane:

  • in the positive -direction
  • in the positive -direction
  • in the negative -direction
  • in the positive -direction

We choose a scale of 1 N:1 cm. Now we sketch the vectors on the Cartesian plane (we can place the vectors anywhere on the Cartesian plane):

Image

Draw the following forces as vectors on the Cartesian plane:

  • in the positive -direction
  • in the negative -direction
  • in the negative -direction
  • in the positive -direction

We choose a scale of 1 N:1 cm. Now we sketch the vectors on the Cartesian plane (we can place the vectors anywhere on the Cartesian plane):

Image

Vectors in two dimensions are not always parallel to an axis. We might know that a force acts at an angle to an axis so we still know the direction of the force and if we know the magnitude we can draw the force vector. For example, we can draw acting at 45° to the positive -direction:

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We always specify the angle as being anti-clockwise from the positive -axis. So if we specified an negative angle we would measure it clockwise from the -axis. For example, acting at −45° to the positive -direction:

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We can use many other ways of specifying the direction of a vector. The direction just needs to be unambiguous. We have used the Cartesian coordinate system and an angle with the -axis so far but there are other common ways of specifying direction that you need to be aware of and comfortable to handle.

Compass directions

We can use compass directions when appropriate to specify the direction of a vector. For example, if we were describing the forces of tectonic plates (the sections of the earth's crust that move) to talk about the forces involved in earthquakes we could talk the force that the moving plates exert on each other.

The four cardinal directions are North, South, East and West when using a compass. They are shown in this figure:

When specifying a direction of a vector using a compass directions are given by name, North or South. If the direction is directly between two directions we can combine the names, for example North-East is half-way between North and East. This can only happen for directions at right angles to each other, you cannot say North-South as it is ambiguous.

Bearings

Another way of using the compass to specify direction in a numerical way is to use bearings. A bearing is an angle, usually measured clockwise from North. Note that this is different to the Cartesian plane where angles are anti- or counter-clockwise from the positive -direction.

The resultant vector

In grade 10 you learnt about adding vectors together in one dimension. The same principle can be applied for vectors in two dimensions. The following examples show addition of vectors. Vectors that are parallel can be shifted to fall on a line. Vectors falling on the same line are called co-linear vectors. To add co-linear vectors we use the tail-to-head method you learnt in Grade 10. In the figure below we remind you of the approach of adding co-linear vectors to get a resultant vector.

In the above figure the blue vectors are in the -direction and the red vectors are in the -direction. The two black vectors represent the resultants of the co-linear vectors graphically.

What we have done is implement the tail-to-head method of vector addition for the vertical set of vectors and the horizontal set of vectors.

Exercise 2:

Find the resultant in the -direction, , and -direction, for the following forces:

  • in the positive -direction
  • in the positive -direction
  • in the negative -direction

We choose a scale of 1 N: 1 cm and for our diagram we will define the positive direction as to the right.

We will start with drawing the vector = 1,5 N pointing in the positive direction. Using our scale of 1 N : 1 cm, the length of the arrow must be 1,5 cm pointing to the right.

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The next vector is = 1,5 N in the same direction as . Using the scale, the arrow should be 1,5 cm long and pointing to the right.

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The next vector is = 2 N in the opposite direction. Using the scale, this arrow should be 2 cm long and point to the left.

Note: We are working in one dimension so this arrow would be drawn on top of the first vectors to the left. This will get confusing so we'll draw it next to the actual line as well to show you what it looks like.

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We have now drawn all the force vectors that are given. The resultant vector is the arrow which starts at the tail of the first vector and ends at the head of the last drawn vector.

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The resultant vector measures 1 cm which, using our scale is equivalent to 1 N and points to the right (or the positive direction). This is . For this set of vectors we have no vectors pointing in the -direction and so we do not need to find .

Find the resultant in the -direction, , and -direction, for the following forces:

  • in the positive -direction
  • in the negative -direction
  • in the positive -direction
  • in the negative -direction

We choose a scale of 1 N: 1 cm and for our diagram we will define the positive direction as to the right.

Before we draw the vectors we note the lengths of the vectors using our scale:


We also note the direction the vectors are in:


We now look at the two vectors in the -direction to find :

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We have now drawn all the force vectors that act in the -direction. To find we note that the resultant vector is the arrow which starts at the tail of the first vector and ends at the head of the last drawn vector in that direction.

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We note that is 1,3 cm or 1,3 N in the positive -direction.

We now look at the two vectors in the -direction to find :

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We have now drawn all the force vectors that act in the -direction. To find we note that the resultant vector is the arrow which starts at the tail of the first vector and ends at the head of the last drawn vector in that direction.

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We note that is 1 cm or 1 N in the negative -direction.

= 1,3 N and points in the positive -direction. = 1 N and points in the negative -direction.

Find the resultant in the -direction, , and -direction, for the following forces:

  • in the positive -direction
  • in the positive -direction
  • in the negative -direction
  • in the positive -direction

We choose a scale of 1 N: 1 cm and for our diagram we will define the positive direction as to the right.

Before we draw the vectors we note the lengths of the vectors using our scale:


We also note the direction the vectors are in:


We now look at the three vectors in the -direction to find :

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We have now drawn all the force vectors that act in the -direction. To find we note that the resultant vector is the arrow which starts at the tail of the first vector and ends at the head of the last drawn vector in that direction.

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We note that is 2 cm or 2 N in the positive -direction.

We now look at the vectors in the -direction to find . We notice that we only have one vector in this direction and so that vector is the resultant.

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We note that is 3 cm or 3 N in the positive -direction.

= 2 N and points in the positive -direction. = 3 N and points in the positive -direction.

Find the resultant in the -direction, , and -direction, for the following forces:

  • in the positive -direction
  • in the negative -direction
  • in the negative -direction
  • in the positive -direction

We choose a scale of 1 cm: 1 N and for our diagram we will define the positive direction as to the right.

Before we draw the vectors we note the lengths of the vectors using our scale:


We also note the direction the vectors are in:


We look at the vectors in the -direction to find . We notice that we only have one vector in this direction and so that vector is the resultant.

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We note that is 2,5 cm or 2,5 N in the negative -direction.

We now look at the three vectors in the -direction to find :

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We have now drawn all the force vectors that act in the -direction. To find we note that the resultant vector is the arrow which starts at the tail of the first vector and ends at the head of the last drawn vector in that direction.

Image

We note that is 3,5 cm or 3,5 N in the positive -direction.

= 2,5 N and points in the negative -direction. = 3,5 N and points in the positive -direction.

Find a force in the -direction, , and -direction, , that you can add to the following forces to make the resultant in the -direction, , and -direction, zero:

  • in the positive -direction
  • in the negative -direction
  • in the negative -direction
  • in the positive -direction

To solve this problem we will draw the vectors on the Cartesian plane and then look at what the resultant vector is. Then we determine what force vector to add so that the resultant vector is 0.

We choose a scale of 1 cm: 1 N and for our diagram we will define the positive direction as to the right.

Before we draw the vectors we note the lengths of the vectors using our scale:


We also note the direction the vectors are in:


We look at the vectors in the -direction to find . We notice that we only have one vector in this direction and so that vector is the resultant.

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We note that is 2,8 cm or 2,8 N in the negative -direction.

So if we add a force of 2,8 N in the positive -direction the resultant will be 0:

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We now look at the three vectors in the -direction to find :

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We have now drawn all the force vectors that act in the -direction. To find we note that the resultant vector is the arrow which starts at the tail of the first vector and ends at the head of the last drawn vector in that direction.

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We note that is 5 cm or 5 N in the positive -direction.

So if we add a force of 5 N in the negative -direction the resultant will be 0:

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We must add a force of 2,8 N in the positive -direction and a force of 5 N in the negative -direction.

Magnitude of the resultant of vectors at right angles

We apply the same principle to vectors that are at right angles or perpendicular to each other.

Sketching tail-to-head method

The tail of the one vector is placed at the head of the other but in two dimensions the vectors may not be co-linear. The approach is to draw all the vectors, one at a time. For the first vector begin at the origin of the Cartesian plane, for the second vector draw it from the head of the first vector. The third vector should be drawn from the head of the second and so on. Each vector is drawn from the head of the vector that preceded it. The order doesn't matter as the resultant will be the same if the order is different.

Let us apply this procedure to two vectors:

  • in the positive -direction
  • in the positive -direction

We first draw a Cartesian plane with the first vector originating at the origin:

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The next step is to take the second vector and draw it from the head of the first vector:

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The resultant, , is the vector connecting the tail of the first vector drawn to the head of the last vector drawn:

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It is important to remember that the order in which we draw the vectors doesn't matter. If we had drawn them in the opposite order we would have the same resultant, . We can repeat the process to demonstrate this:

We first draw a Cartesian plane with the second vector originating at the origin:

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The next step is to take the other vector and draw it from the head of the vector we have already drawn:

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The resultant, , is the vector connecting the tail of the first vector drawn to the head of the last vector drawn (the vector from the start point to the end point):

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Exercise 3:

Sketch the resultant of the following force vectors using the tail-to-head method:

  • in the positive -direction
  • in the negative -direction

We first draw a Cartesian plane with the first vector originating at the origin:

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The next step is to take the second vector and draw it from the head of the first vector:

Image

The resultant, , is the vector connecting the tail of the first vector drawn to the head of the last vector drawn:

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Sketch the resultant of the following force vectors using the tail-to-head method:

  • in the positive -direction
  • in the positive -direction
  • in the negative -direction
  • in the negative -direction

We first sketch the vectors on the Cartesian plane. We choose a scale of 1 cm = 2 N. Remember to sketch starting at the head of , starting at the head of and starting at the head of .

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The resultant, , is the vector connecting the tail of the first vector drawn to the head of the last vector drawn:

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Sketch the resultant of the following force vectors using the tail-to-head method by first determining the resultant in the - and -directions:

  • in the positive -direction
  • in the negative -direction
  • in the negative -direction
  • in the negative -direction

We first determine

Draw the Cartesian plane with the vectors in the -direction:

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This is since it is the only vector in the -direction.

Secondly determine

Next we draw the Cartesian plane with the vectors in the -direction:

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Now we draw the resultant vectors, and head-to-tail:

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You can check this answer by using the tail-to-head method without first determining the resultant in the -direction and the -direction.

Sketch the resultant of the following force vectors using the tail-to-head method by first determining the resultant in the - and -directions:

  • in the positive -direction
  • in the negative -direction
  • in the negative -direction
  • in the negative -direction

We choose a scale of 1 cm : 2 N.

We first determine

Draw the Cartesian plane with the vectors in the -direction:

Image

This is since it is the only vector in the -direction.

Secondly determine

Next we draw the Cartesian plane with the vectors in the -direction:

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Now we draw the resultant vectors, and head-to-tail:

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You can check this answer by using the tail-to-head method without first determining the resultant in the -direction and the -direction.

Sketching tail-to-tail method

In this method we draw the two vectors with their tails on the origin. Then we draw a line parallel to the first vector from the head of the second vector and vice versa. Where the parallel lines intersect is the head of the resultant vector that will also start at the origin. We will only deal with perpendicular vectors but this procedure works for any vectors.

Interesting Fact:

When dealing with more than two vectors the procedure is repetitive. First find the resultant of any two of the vectors to be added. Then use the same method to add the resultant from the first two vectors with a third vector. This new resultant is then added to the fourth vector and so on, until there are no more vectors to be added.

Let us apply this procedure to the same two vectors we used to illustrate the head-to-tail method:

  • in the positive -direction
  • in the positive -direction

We first draw a Cartesian plane with the first vector originating at the origin:

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Then we add the second vector but also originating from the origin so that the vectors are drawn tail-to-tail:

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Now we draw a line parallel to from the head of :

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Next we draw a line parallel to from the head of :

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Where the two lines intersect is the head of the resultant vector which will originate at the origin so:

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You might be asking what you would do if you had more than 2 vectors to add together. In this case all you need to do is first determine by adding all the vectors that are parallel to the -direction and by adding all the vectors that are parallel to the -direction. Then you use the tail-to-tail method to find the resultant of and .

Exercise 4:

Sketch the resultant of the following force vectors using the tail-to-tail method:

  • in the positive -direction
  • in the negative -direction

We first draw a Cartesian plane with the first vector originating at the origin:

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Then we add the second vector but also originating from the origin so that the vectors are drawn tail-to-tail:

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Now we draw a line parallel to from the head of :

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Next we draw a line parallel to from the head of :

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Where the two lines intersect is the head of the resultant vector which will originate at the origin so:

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Sketch the resultant of the following force vectors using the tail-to-tail method by first determining the resultant in the - and -directions:

  • in the positive -direction
  • in the negative -direction
  • in the negative -direction
  • in the negative -direction

We need to determine and and then use these to find the resultant.

Determine .

Draw the Cartesian plane with the vectors in the -direction:

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This is since it is the only vector in the -direction.

Secondly determine

Next we draw the Cartesian plane with the vectors in the -direction:

Image

Now we draw the resultant vectors, and tail-to-tail:

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Now we can draw the lines to show us where the head of the resultant must be:

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And finally we find the resultant:

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Sketch the resultant of the following force vectors using the tail-to-tail method by first determining the resultant in the - and -directions:

  • in the positive -direction
  • in the negative -direction
  • in the negative -direction
  • in the negative -direction

We choose a scale of 1 cm = 2 N.

We first determine

Draw the Cartesian plane with the vectors in the -direction:

Image

This is since it is the only vector in the -direction.

Secondly determine

Next we draw the Cartesian plane with the vectors in the -direction:

Image

Now we draw the resultant vectors, and tail-to-tail:

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Now we can draw the lines to show us where the head of the resultant must be:

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And finally we find the resultant:

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Closed vector diagrams

A closed vector diagram is a set of vectors drawn on the Cartesian using the tail-to-head method and that has a resultant with a magnitude of zero. This means that if the first vector starts at the origin the last vector drawn must end at the origin. The vectors form a closed polygon, no matter how many of them are drawn. Here are a few examples of closed vector diagrams:

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In this case there were 3 force vectors. When drawn tail-to-head with the first force starting at the origin the last force drawn ends at the origin. The resultant would have a magnitude of zero. The resultant is drawn from the tail of the first vector to the head of the final vector. In the diagram below there are 4 vectors that also form a closed vector diagram.

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In this case with 4 vectors, the shape is a 4-sided polygon. Any polygon made up of vectors drawn tail-to-head will be a closed vector diagram because a polygon has no gaps.

Using Pythagoras' theorem to find magnitude

If we wanted to know the resultant of the three blue vectors and the three red vectors in Figure Figure 5 we can use the resultant vectors in the - and -directions to determine this.

The black arrow represents the resultant of the vectors and . We can find the magnitude of this vector using the theorem of Pythagoras because the three vectors form a right angle triangle. If we had drawn the vectors to scale we would be able to measure the magnitude of the resultant as well.

What we've actually sketched out already is our approach to finding the resultant of many vectors using components so remember this example when we get there a little later.

Note: we did not determine the resultant vector in the worked example above because we only determined the magnitude. A vector needs a magnitude and a direction. We did not determine the direction of the resultant vector.

Graphical methods

Graphical techniques

In grade 10 you learnt how to add vectors in one dimension graphically.

We can expand these ideas to include vectors in two-dimensions. The following worked example shows this.

In the case where you have to find the resultant of more than two vectors first apply the tail-to-head method to all the vectors parallel to the one axis and then all the vectors parallel to the other axis. For example, you would first calculate from all the vectors parallel to the -axis and then from all the vectors parallel to the -axis. After that you apply the same procedure as in the previous worked example to the get the final resultant.

Algebraic methods

Algebraic addition and subtraction of vectors

In grade 10 you learnt about addition and subtraction of vectors in one dimension. The following worked example provides a refresher of the concepts.

We can now expand on this work to include vectors in two dimensions.

Direction

For two dimensional vectors we have only covered finding the magnitude of vectors algebraically. We also need to know the direction. For vectors in one dimension this was simple. We chose a positive direction and then the resultant was either in the positive or in the negative direction. In grade 10 you learnt about the different ways to specify direction. We will now look at using trigonometry to determine the direction of the resultant vector.

We can use simple trigonometric identities to calculate the direction. We can calculate the direction of the resultant in the previous worked example.

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