Subtraction of vectors graphic method, examples, exercises

The vector subtraction or vector subtraction between vectors or Y v denoted by or - v, is calculated by adding the vector or with the vector opposite v. Algebraically the subtraction is expressed as follows:

or - v = or + (-v)

It is possible to carry out vector subtraction following various procedures, for example in graphical form, in this way a vector v is drawn by an oriented line segment - an arrow-.

The length of the arrow corresponds to the module of the vector, the inclination - with respect to a given reference line - indicates the direction and the end indicates the direction of the vector.

The vector opposite v it has the same length and direction, but the opposite direction. Then, before doing the subtraction between or Y v, it is necessary to draw the vector opposite v, and add this vector to u.

It is very important to note that vector subtraction is not commutative, that is, the order of the vectors does alter the result, therefore:

or - v ≠ v - or

The graphical procedure can be carried out using any of these methods, the steps of which we will explain below:

-Triangle method.

-Parallelogram method.

Article index

• 1 Graphical method of vector subtraction
  • 1.1 Triangle method
  • 1.2 Parallelogram method
• 2 Examples of vector subtraction
  • 2.1 - Example 1
  • 2.2 - Example 2
• 3 Exercise resolved
  • 3.1 Solution
• 4 References

Graphical vector subtraction method

Triangle method

In figure 1 we have the first of the methods to graphically subtract two vectors. Its about triangle method, because the figure that is formed by establishing the vectors is a triangle, as we can see in the left image.

To do the subtraction or - v we proceed as follows:

-Draw the vector -v from vector v, by translation with a ruler and square, but changing the direction of the arrow (left image).

-Moves to vector -v in such a way that its origin coincides with the end of the vector or (right image).

-Next a vector is drawn (in red in the right image) that goes from the origin of or to the end of v. Call D y is the difference vector:

D = or - v

Parallelogram method

In the parallelogram method, the vectors to be added or subtracted must coincide at their points of origin. Suppose we want to find or - v With our vectors shown above, the steps to find the subtraction of vectors by this method are as follows:

-Determine the vector opposite v, What is it -v, as described above for the triangle method.

-Carefully Translate Vectors or Y -v in such a way that their origins coincide.

-Now segmented parallel lines are drawn starting from the ends of each vector. The figure that is formed is a parallelogram and in special cases in which the vectors are perpendicular, a rectangle or a square results..

-Finally, a vector is drawn that starts from the common origin of or Y v to the extreme where the segmented parallel lines intersect. This is the vector D or subtraction.

Important

Another way to do the subtraction is to draw the parallelogram as if you wanted to add the vectors.

But instead of drawing the usual diagonal of the sum, which goes from the common origin to the intersection of the parallels, the opposite or shorter diagonal, as seen in the figure:

Examples of vector subtraction

- Example 1

A ship sails on a river and does so in the opposite direction to the current. An observer on land observes that the speed of the boat is reduced due to the action of the current.

The speed is a vector and in this example, the speed of the boat is pointing in one direction and the speed of the current has the same direction and opposite direction. The net speed of the ship is the sum of both vectors.

For example, if the instruments of the boat indicate that it is moving at v '= + 40 km / h and an observer on the shore measures that the boat is moving at v = + 30 km / h. Since v = v '+ Vc, where Vc is the speed of the current that is calculated by subtracting the speeds v and v' respectively: Vc = v - v '= 30 km / h - 40 km / h = -10 km / h.

- Example 2

In kinematics we have important vectors that describe changes:

-Offset for changes in position.

-Average velocity, to quantify how fast the position varies over time.

-Acceleration, for speed changes as a function of time.

The displacement vector

The displacement vector describes the change in position that a body experiences during its movement.

Let's see for example a particle that describes the plane path shown in the figure, in which it passes from point P₁ to point P_two.

The vectors directed from the origin of the x-y coordinate system to these points are the position vectors r₁ Y r_two, while the displacement vector is Δr, that goes from P₁ to P_two. It is true that:

Δr = r_two - r₁

Therefore, the displacement vector is the subtraction between the final position vector and the initial position vector, as shown in the following figure. Its units are also those of position: meters, feet, miles, centimeters and more..

The mean velocity and mean acceleration vectors

For its part, the mean velocity vector v_m is defined as the offset multiplied by the inverse of the time interval:

Exercise resolved

A particle that describes a circle takes 5 s to pass from point A to point B. At A it has a velocity v_TO = 60 km / h towards the + x axis and at B is v_B = 60 km / h towards + y. Determine its average acceleration graphically and analytically.

Solution

In graphical form, direction and the direction of the mean acceleration are determined by:

In the following image is the subtraction v_B - v_TO, using the triangle method, since the average acceleration to_m is proportional to Δv. The triangle formed has the two legs equal and therefore the acute internal angles measure 45º each..

Analytically, if the + x direction coincides with the unit vector i and the + y direction with the unit vector j, then:

Δv = 60 km / h j - 60 km / h i

Taking Δt = 5 s, according to the information in the statement, the mean acceleration is:

to_m = (60 km / h j - 60 km / h i) / 5 s = 12 (j-i) km / (h.s)

References

1. Bauer, W. 2011. Physics for Engineering and Sciences. Volume 1. Mc Graw Hill.
2. Bedford, 2000. A. Engineering Mechanics: Statics. Addison wesley.
3. Figueroa, D. (2005). Series: Physics for Science and Engineering. Volume 1. Kinematics. Edited by Douglas Figueroa (USB).
4. Giambattista, A. 2010. Physics. 2nd. Ed. McGraw Hill.
5. Sears, Zemansky. 2016. University Physics with Modern Physics. 14th. Ed. Volume 1.
6. Tipler, P. 2006. Physics for Science and Technology. 5th Ed. Volume 1. Editorial Reverté.