I.5.2. Composition of collinear vectors
To compose vectors we have two cases:
- Collinear vectors.
- Nonlinear 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.
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.
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)
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.
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).
