I.5. 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.5.1. Composition of collinear vectors.
Collinear vectors are vectors that have the same direction.
1. 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.
🔦 Remark
It is similar to the algebraic addition of numbers with the same sign (numbers are added and the common sign is passed to the result).
2. 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.
🔦 Remark
It is similar to the algebraic addition of numbers with different signs (the numbers are subtracted, the one with the higher value minus the one with the lower value and the result is passed the sign of the higher number)
🔦 Remark
In mathematics, you learned that a two-dimensional Cartesian coordinate system is usually defined by two axes at right angles to each other, forming a plain. The horizontal axis is normally labeled Ox, and the vertical axis is denoted by Oy. The point of intersection of the axes is called origin and is denoted by O. To specify a certain point on a two-dimensional coordinate system, first indicate the unit x (abscissa), followed by the unit y (ordinate).
Conventionally, the intersection of the two axes gives rise to four regions, called quadrants, denoted by the Roman numerals I (+, +), II (-, +), III (-, -) and IV (+, -) . In the first quadrant, both coordinates are positive, in the second quadrant the abscissas are negative and the ordinates are positive, in the third quadrant both coordinates are negative and in the fourth quadrant, the abscissas are positive and the ordinates are negative.
🔓 Solved problems
1. Two children pull a sledge on a horizontal road, to the west, with the forces F1 = 400 N, respectively F2 = 800 N. Compose the two forces of children.
Solution:
The resulting vector has:
- numerical value equal to the sum of the numerical values of the component vectors, ie F = F1 + F2 = 400 N + 800 N = 1200 N.
- direction common with component vectors: horizontal.
- way common with component vectors: to the left.
To represent the resulting vector we must choose an appropriate standard so that we have enough place for the drawing on the notebook page.
Standard: 1 cm: 200 N
The resulting vector segment is 1200 : 200 = 6 cm.
2. Two forces act on the spring of a dynamometer suspended from a support, one of 60 N, vertically downwards, the other of 150 N, vertically upwards. What force does the dynamometer indicate?
Solution:
The resulting vector has:
- numerical value equal to the difference of the numerical values of the component vectors, ie F = F2 - F1 = 150 N - 60 N = 90 N.
- direction common with component vectors: vertical.
- way of the higher value vector: up.
To represent the resulting vector we must choose an appropriate standard so that we have enough place for the drawing on the notebook page.
Standard: 1 cm: 30 N.
The resulting vector segment is 90:30 = 3 cm.
3. Two forces act on a dynamometer, one of 150 N in a vertical direction, upward direction. The dynamometer indicates a force of 90 N, its spring being elongated vertically downwards. Draw the second force acting on the dynamometer spring.
Solution:
Standard: 1 cm : 60 N.
We write the vectorial equation:
We write the scalar equation taking into account the sign convention:
-90 N = 150 N + F2 (F is taken with minus, because it is vertically down, and F1 is taken with plus, because it is vertically up)
F2 = -90 N -150 N = -240 N. It turns out that F2 has a segment of 240 : 60 = 4 cm, in a vertical direction, the downward way (because it gave us the minus sign).
I.5.2. Composition of nonlinear vectors.
Nonlinear vectors are vectors that do not have the same direction.
The composition of nonlinear vectors (which do not have the same direction) is done according to two rules: the parallelogram rule and the polygon rule.
I.5.2.1. Parallelogram rule.
Parallelogram rule is used for the addition of two nonlinear vectors which have the same point of application, 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.
🔓 Solved problem
1. A river flows east at a speed of 60 km/h. A boat goes on the river in its direction of flow at a speed of 100 km/h, in a direction that makes an angle of 30° to the river bank. What is the speed of the boat towards the shore? Graphs using the following scale: 1 cm: 20 km/h.
Solution:
v1 = 60 km/h, horizontal direction, to the right
v2 = 100 km/h, direction that makes an angle of 30° with the horizontal, upwards.
Standard: 1cm : 20 km/h.
I.5.2.2. Polygon rule.
Polygon rule is used for adding of several nonlinear vectors which do not have the same point of application, 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.
🔓 Solved problem
1. A cyclist goes 20 km East, then 40 km South, then 80 km West and 60 km North. Determine the resulting vector, ie how far the cyclist is compared with th initial point of departure (0).
Solution:
d1 = 20 km, horizontal direction, to the right
d2 = 40 km, vertical direction, down
d3 = 80 km, horizontal direction, to the left
d4 = 60 km, vertical direction, upwards
Standard: 1 cm : 10 km.
-
We represent the first movement vector d1. We put the second vector at the top of the first vector, and so on, until we represent all four vectors.
-
The resulting vector is the segment obtained by joining the origin (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 standard: d = 6,3 ∙ 10 = 63 km. So the cyclist is at a distance of 63 km from the starting point, after the whole race.
