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Vector Diagram

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Jetzt kostenlos anmeldenThe distance from London to Paris is around \( 350\,\mathrm{km} \) if you flew straight there in a plane. However, if you flew the same distance in a different direction then you would end up somewhere else, of course. Amsterdam is also around \( 350\,\mathrm{km} \) from London. To get from one point to another you must know not just the distance but also the direction. In physics, quantities with both a size and a direction are called vectors. In this article, we will learn the different ways of drawing vectors on vector diagrams and how these diagrams can be used.

Vector diagrams are extremely important in physics and you will come across them throughout your studies. They are defined as follows:

A **vector diagram** is a diagram that shows the relative magnitudes and directions of vector quantities.

As expected from the name, a vector diagram is a way to visualise vectors. It is important to understand what a vector quantity is exactly:

A **vector quantity **has both a magnitude and a direction.

The reason for giving the definition of a vector quantity is to distinguish it from a scalar quantity, which only has a magnitude and does not have a direction. Some examples of different scalar and vector quantities are given in the table below.

Scalars | Vectors |

Mass | Force |

Speed | Velocity |

Distance | Displacement |

Time | Acceleration |

Energy | Momentum |

Table 1. Examples of different scalar and vector quantities. |

Now that we have defined a vector diagram, let us take an example of one to see what these diagrams represent. Figure 1 shows the forces acting on a skydiver. They feel a downward force due to gravity and an upward force due to air resistance. In the diagram, the length of the arrows shows the relative magnitudes of the forces.

Vector diagrams are useful in physics because they can be used to find the magnitude and direction of the overall vector due to multiple vectors added together, which is called the resultant vector.

A **resultant vector** is the sum of two or more vectors.

A resultant vector can be found by using **vector addition**. This involves placing the vector arrows one after the other, from tip to tail, while maintaining their directions. Afterwards, an arrow can be made from the start of the first vector to the end of the last vector to find the resultant vector. In Figure 2, we see the addition of \(V_1\), shown as a blue arrow, with \(V_2\), shown as a green arrow, forming the resultant vector \(V_R\) in red. This is done by placing \(V_1\) and \(V_2\) tip to tail and then drawing the resultant vector from the tail of \(V_1\) to the tip of \(V_2\).

Vector diagrams normally use a scale to **help represent the vectors' magnitudes**. If you are given several vectors of different magnitudes, then in order to find a scale to represent them, you could take the ratio of the largest to the smallest magnitude and use this value as the unit length for representing the rest of the vectors' magnitudes. This is just an example - there are different ways of arriving at a scale for representing vectors.

Consider drawing a vector diagram for someone who walks \( 4\,\mathrm{km} \) East followed by \( 3\,\mathrm{km} \) North. If you used a ruler and piece of paper, then a suitable scale to use would be to set \( 1\,\mathrm{km} \) equal to \( 1\,\mathrm{cm} \) on the paper. You could then draw a resultant vector from the starting point to the endpoint and should find that the total displacement of the person was \( 5\,\mathrm{km} \) in the North-East direction.

Drawing a vector diagram is relatively simple. First, you must **define a scale for the magnitudes of the vectors**, as discussed above. If you were using a ruler and drawing on a piece of paper, then it might be suitable to use the \( \mathrm{cm} \) markings on your ruler for the scale. For example, you could set \( 1\,\mathrm{cm} \) to \( 1\,\mathrm N \) when dealing with forces, or maybe \( 1\,\mathrm{cm} \) to \( 1\,\mathrm m/\mathrm s \) when working with velocities. Of course, the scale depends on the magnitudes of the vectors being used.

Then you need to decide on a reference direction. Once you have these two decisions made, you can adapt the real length of your vectors to the scale and represent them by taking into account their direction with respect to the reference direction. Importantly, you always have to remember to draw the vector as an arrow, with the arrowhead pointing in the direction of the vector.

There are different types of vector diagrams. Sometimes it is more useful to draw the vectors from tip to tail, so that the resultant vector can be calculated easily, such as for displacements and velocities. In other situations, it is better to draw the vectors coming from one point. For example, when drawing vector diagrams for forces, the vectors are drawn from where they act on the object so that their effects on the object can be visualized.

The most common types of vector diagrams involve forces. These diagrams are called free-body diagrams.

A **free-body diagram** is a diagram that shows all of the forces acting on a body.

A general free-body diagram is shown in figure 4. The arrows on a free-body diagram originate from the centre of mass, as this is the point where all forces can be presumed to act.

The **centre of mass** of an object is the point at which all of the mass of the object can be considered to be.

Free-body diagrams can be used to find the acceleration of an object due to the forces acting on it. Newton's second law states that

$$\vec F=m\vec a$$

where \( \vec F \) is the resultant force acting on an object in \( \mathrm N \), \( m \) is the mass of the object in \( \mathrm{kg} \) and \( \vec a \) is the acceleration of the object in \( \mathrm m/\mathrm{s^2} \). This means that if the resultant force acting on a body is known, then its acceleration can be found by rearranging Newton's second law to

$$\vec a=\frac{\vec F}{m}.$$

If the direction of the vector is less important, such as when considering an object's free-fall, then we can use the vector magnitude form of the equation.

$$a=\frac{F}{m}$$

A car accelerates due to a constant driving force of \( 1000\,\mathrm N \). The frictional force as the car moves along the ground is \( 200\,\mathrm N \). Ignoring air resistance, what is the acceleration of the car? The car's mass is \( 1000\,\mathrm{kg} \).

The magnitude of the resultant force acting on the car will be equal to the driving force minus the frictional force:

$$\begin{align}F&=F_D-F_F\\&=1000\,\mathrm N-200\,\mathrm N\\&=800\,\mathrm N\end{align}$$

The acceleration can be found from the rearranged form of Newton's law, which is

$$a=\frac Fm,$$

where the quantities have been stated as magnitudes as the car is moving in a straight line, so a positive result will correspond to an acceleration in the forward direction and a negative result to the backward direction. The mass of the car is given in the question as \( 1000\,\mathrm{kg} \) so the acceleration of the car is equal to

$$a=\frac{800\,\mathrm N}{1000\,\mathrm{kg}}=0.8\,\mathrm m/\mathrm{s^2}.$$

Vector diagrams can be useful when answering practice questions.

A helicopter flies \( 10\,\mathrm{km} \) west before directly flying \( 6\,\mathrm{km} \) south. Finally, it flies \( 18\,\mathrm{km} \) east. What is the total distance of its final destination from its starting point?

The total distance to the final destination can be calculated by finding the magnitude of the resultant vector found from adding the displacement vectors representing each leg of the trip. A vector diagram for this is shown in figure 5. The total distance is \( 10\,\mathrm{km} \).

Vectors can be split up into **components**. In three dimensions, three perpendicular axes can be drawn and the resultant velocity vector can be decomposed into three velocity vectors in each direction. This is shown in figure 6. Work through the following example to gain a better understanding of this concept.

An American football player throws the ball, hoping to find her teammate. Take the \( x \) direction parallel to the field in line with the path of the ball and the \( y \) axis perpendicular to the field. As it leaves her hand, the ball has a velocity in the \( x \) direction of \( 24\,\mathrm m/\mathrm s \) and a velocity in the \( y \) direction of \( 7\,\mathrm m/\mathrm s \). What is the initial *speed *of the ball?

The magnitude of a velocity vector is equal to the speed.

One way of finding the speed of the ball is by drawing a vector diagram and measuring the magnitude of the resultant vector. Using the values given in the question, the vector diagram in figure 7 can be drawn.

You should be able to find the speed of the ball as \( 25\,\mathrm m/\mathrm s \). The speed could also have been found by using Pythagoras' theorem The two velocity components could be taken as the two perpendicular sides of a right-angled triangle.

- A vector diagram is a diagram that shows the relative magnitudes and directions of vector quantities.
- A vector quantity
- A scalar quantity only has a magnitude.
- A resultant vector is the sum of two or more vectors.
- Vector diagrams normally use a scale to help represent the vectors' magnitudes.
- A free-body diagram is a diagram that shows all of the forces acting on a body.
- The centre of mass of an object is the point at which all of the mass of the object can be considered to be.
- Newton's second law states that \( F=ma \).

- Fig. 1 - Flight paths to Europe, StudySmarter.
- Fig. 2 - Skydiver force diagram, StudySmarter Originals.
- Fig. 3 - Displacement vectors, StudySmarter Originals.
- Fig. 4 - Free-body diagram, StudySmarter Originals.
- Fig. 5 - Addition of multiple vectors, StudySmarter Originals
- Fig. 6 - Velocity components, StudySmarter Originals
- Fig. 7 - Vector components example, StudySmarter Originals

**vector diagram** is a diagram that shows the relative magnitudes and directions of vector quantities.

To draw a vector diagram you must define a scale for the magnitudes of the vectors. Then you need to decide on a reference direction.

Once you have these two decisions made, you can adapt the real length of your vectors to the scale and represent them by taking into account their direction with respect to the reference direction. Importantly, you always have to remember to draw the vector as an arrow, with the arrowhead pointing in the direction of the vector.

**vector diagram** is a diagram that shows the relative magnitudes and directions of vector quantities.

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