I.7. Summary - Vectors. Vector composition.

📚 Summary - Vectors. Vector composition.

Scalar physical parameter (in short, scalar) is the physical parameter that is completely characterized by:

• Numeric value;
• Measurement unit.

The vast majority of physical parameters are scalar: length, area, volume, time, mass, density, temperature, current intensity, electrical voltage etc.

Vectorial physical parameter (in short, vector) is the quantity that is completely characterized by:

• Numeric value (module);
• Measurement Unit;
• Orientation:
  • Direction;
  • Sens.

To differentiate a scalar from a vector, an arrow is placed above the symbol of the vectorial physical parameter.

Examples of vectorial physical parameters:

Graphical representation of a vector..

To graph a vector we must draw:

• A point of application, called the origin of the vector and denoted by 0;
• A line that gives the direction of the vector;
• The segment of the vector that we measure with the ruler using a standard (the value of the vector is divided by the standard chosen to find out the length of the segment);
• An arrow at the end of the segment that will indicate the direction of the vector.

Addition (composition) of vectors.

Adding two (F₁ and F₂) or more vectors means determining the resulting vector (F).

The vectorial equation is:

To compose vectors we have two cases:

I. Composition of collinear vectors (that have the same direction).

A. If the collinear vectors have the same meaning (the angle between them is 0°) then the resulting vector has:

• numerical value equal to the sum of the numerical values of the component vectors
• direction common with component vectors.
• way common with component vectors.

B. If the collinear vectors have opposite meanings (the angle between them is 180°) then the resulting vector has:

• numerical value equal to the difference of the numerical values of the component vectors (subtract from the one with the higher value the one with the lower value)
• direction common with component vectors
• way of the higher value vector.

Conventionally, the intersection of the two axes gives rise to four regions, called quadrants, denoted by the Roman numerals I (+, +), II (-, +), III (-, -) and IV (+, -) .

II. The composition of nonlinear vectors (which do not have the same direction) is done according to two rules:

II.1. For the addition of two nonlinear vectors, which have the same point of application is used the Parallelogram rule by going through the following four steps:

• Draw the two vectors so that they have the same point of application.
• With the segments of the 2 vectors, a parallelogram is formed (quadrilateral with parallel and equal opposite sides).
• Draw the diagonal of the parallelogram that has a common point with the two vectors. This segment represents the resulting vector, which is noted and an arrow is placed at the end.
• With the ruler we measure the segment of the resulting vector and with the simple rule of three, we find its numerical value.

II.2 For the addition of several nonlinear vectors, which do not have the same point of application is used the Polygon rule by going through the following steps:

• Draw the first vector.
• The second vector is drawn with the origin at the top of the first vector, keeping its direction.
• The third vector is drawn with the origin at the top of the second vector, keeping its direction and so on, until we represent all vectors.
• The resulting vector is the segment that is obtained by joining the origin of the first vector (0) with the vertex of the last vector. The resulting vector has the same vertex as the last vector vertex.
• The value of the resulting vector is obtained by measuring its segment with the ruler and then multiplying by the given (chosen) standard.

📝 Self-assessment sheet - Vector composition - Variant I

1. Two forces act on a body. Known: F₂ = 60 N, acting vertically downwards and the resulting force, F = 40 N, acting vertically upwards. Graph the force F₁ using the following standard: 1cm : 20N. -1,5p

2. Compose the following two vectors, which have the same application point: -2p

F₁ = 2000 N, direction that makes an angle of 60° with the horizontal, down

F₂ = 2400 N, horizontal direction, to the left

Standard: 1 cm : 400 N

3. A cyclist goes 150 km to the east, then 60 km to the north, then 120 km to the west and finally 180 km to the south. Use as standard 1 cm : 30 km. Determine the resulting vector. -1,5p

&#128221 Self-assessment sheet - Vector composition - Variant II

1. Two forces act on a body. Known: F₁ = 2500 N, acting horizontally, to the left and the resulting force, F = 1500 N, horizontally, to the left. Graph the force F₂, using the following standard: 1 cm : 500N -1,5p

2. Compose the following two vectors, which have the same application point: -2p

F₁ = 50 N, direction that makes an angle of 50° with the vertical, upwards

F₂ = 70 N,vertical direction, upwards

Standard: 1 cm : 10 N

3. A pedestrian goes South 240 m, then East 200 m, then North 120 m and finally West 160 m. Standard 1 cm : 40 m. Determine the resulting vector. -1,5p

Solution: