Last revision 091412
Vectors and vector addition
Displacement, force, velocity and acceleration are all vectors, that is, they all have magnitude and direction. Magnitude is the numerical part: 5 meters, 17 newtons, 20 meters per second. Direction can be simply up, down, left or right, or direction can be north, south, east or west or some variation like southeast, or it can be a heading on a compass, like 135 degrees (which is also southeast). Sometimes we will be dealing with more than one vector quantity acting on an object at a time. For instance, we know that forces come in pairs, so we know that at least two forces act on an object, and what if we have several forces acting on an object? What will be the net effect of the combination of forces? This requires that we be able to add vectors. Adding vectors is more than adding the magnitudes, we must also take into account the direction of each vector. A vector, whether it is a force, a displacement or a velocity or an acceleration, can be represented by an arrow with the length of the arrow being proportional to the magnitude and the direction of the arrow representing the direction of the vector. 20N up 2 km northeast 5 m right 35 m/s at 240o Adding vectors head to tail The simplest representation of vector addition is to add them "head to tail". The arrowhead is the head of a vector, and you get the idea. Examples of placing vectors head to tail. We often use a x-y grid as a reference for the vectors. The first vector that we draw has its tail at the origin of the x-y axes.
The vector sum has been added below, and is shown in red. The vector sum, which is called the resultant, goes from the tail of the first vector to the head of the last vector.
Just like 3 + 5 + 2 = 10 and 5 + 2 + 3 = 10, it makes no difference in which order we add the vectors head to tail. The sum, the resultant, will always be the same. We start with the first vector at the origin and add them in different orders. Look at the magnitude and direction of the resultant in each case.
Interative simulation Run the PhET simulation called Vector Addition as shown below and experiment with adding vectors head to tail in different orders. Start with two vectors and then add more. Turn component display to "none" and click on "Show sum".
Adding vectors with the parallelogram method A parallelogram is a four-sided figure in which opposites sides are parallel and the same length. A square is a prallelogram and so is a rectangle. And so is this: Unlike the head-to-tail method of adding vectors, the "parallelogram method" allows us to only add two vectors at a time. In the parallelogram method place the two vectors so that the tails are at the same point and then using the two vectors as two of the sides, complete the parallogram. Step one - draw the two vectors on the x and y axes with the tails together. The lengths should be proportional to the magnitude of the vector.
Step two - complete the parallelogram.
Step three - Draw the diagonal of the parallelogram. This is the resultant, the sum of the two vectors. The diagonal represents the magnitude and direction of the vector sum. Notice that all three tails are at the same point.
The head-to-tail method gives a good graphical representation of the sum of the two or more vectors, but it isn't well suited to numerical answers. The parallelogram method improves on this by allowing us to compute the length of the resultant and its direction. The downside is that in order to compute the magnitude (side c) and the direction of the resultant (based on angle B) we must solve a non-right triangle. This requires us to use the law of cosines and the law of sines. Law of cosines: c = a 2 + b 2 − 2ab cos C
Law of sines:
resultant (side c) C = 115o B side a = 10.0 m
a b c = = sin A sin B sin C
The law of cosines is interesting because when angle C is 90 degrees, as in a right triangle, the cosine of 90 is zero and the entire (2ab cos C) is zero and the equation becomes the Pythagorean theory. The parallelogram method is much easier to use when the vectors are at right angles and the triangles to solve are right triangles. This brings us the next method for doing vector addition.
side b = 7.0 m
But first a word about directions To determine a numerical value for the direction of the resultant, we must deal with some angles. There are two common ways to assign numerical values to angles, one of which you are probably familiar with from math classes. The second you may have seen if you are a Boy Scout or play combat games. The second ways deals with a compass. When the word "heading" is used it will probably refer to directions on a compass. Regardless of which numbering system we use, the directions are assigned positive and negative values as shown in the third diagram. signs to apply to most often seen in math class
90
"compass rose"
180
0
90 E
270 W
270
vector components
0N
180 S
+
-
+
-
Resolving vectors into components Another way to add vectors is the "component method". One of its steps actually includes the parallelogram method, but the advantage is that all of the vectors which are being added are at right angles to each other. A second advantage of the component method is that we can add more than two vectors at the same time. Instead of adding the vectors together directly, in the component method we are adding the components of the vectors. Every vector can be resolved into two mutually perpendicular vectors called "components". Usually the components lie along the x and y axes.
7.0 m 25
o
Our work is simplified because one of the vectors is already on the x-axis. Therefore, it has an x-component of 10.0 m in the positive direction, and it has a y-component of zero. The 7.0 m vector on the other hand must be resolved into its two perpendicular components.
10.0 m
7.0 m 25
o
Imagine that there are two vectors (the components) which lie along the x and y axes that when added together equal the 7.0 m vector at a heading of 25 degrees. We can visualize the components of the vector by imagining a flashlight shining down onto the vector and seeing the shadow that it casts onto the x-axis. Do the same with a flashlight to the right so that a shadow is cast onto the y-axis.
opp = ? We must solve the right triangle with a 25 degree angle and a hypotenuse of 7.0 m.
7.0 m adj = ?
25o
25o
7.0 m
This is the x-component. It is the same length as the opposite side. opp = ?
adj hyp
adj = ?
cos θ =
adj = hyp × cos θ
sin θ = 25o
opp hyp
7.0 m
opp = hyp × sin θ
adj = 7 × cos 25 = 6.34 m
opp = 7 × sin 25 = 2.96 m
Normally, we would do this for each vector that is to be added. Next, we add up the x and y components in order to get the components of the resultant. The x-component of the resultant is the sum of all of the x-components, and the y-component of the resultant is the sum of all of the y-components. Here is where we need to pay attention to the signs of the components. All of the components are in positive directions. The x-component of the resultant is the sum of the x-components: xR = x1 + x2 = (+10.0 m) + (+2.96 m) = +12.96 m
What we know: x1 = +10.0 m y1 = 0 x2 = +2.96 m y2 = + 6.34 m
6.34 m
The y-component of the resultant is the sum of the y-components: yR = y1 + y2 = (0) + (+6.34 m) = +6.34 m Pick either of the triangles and use the Pythagorean theorem to find the length of the diagonal of the parallelogram … the resultant. Use the tangent to determine the heading of the resultant.
?m
θ
R = 12.96 2 + 6.34 2 = 14 .4 m
12.96 m
θ = arctan
opp 12.96 = arctan = 63.9 o adj 6.34
resultant (side c) C = 115o B side a = 10.0 m
side b = 7.0 m
Notice how the resultant is the same, regardless of the method.
Return to the PhET simulation called Vector Addition to investigate vectors and their components and the adding of vectors by the component method. Use the simulation to explore resolving vectors into their components. You will see the vector, the direction and the components in the box at the top. Notice how the components have negative values for the vector below. Calculate the magnitudes of the components and compare them to the values given in the boxes.
The simulation below has two vectors added head to tail. Their components are also shown, as is the vector sum. Notice how the components add to give the components of the resultant. Click on the vector or the sum to see the values in the boxes. Do the calculations for each of the two vectors and add the components to get the components of the resultant. Determine the magnitude and the direction of the resultant.
