EVALUACIÓN DEL DESEMPEÑO PROFESIONAL DE LOS DIRECTIVOS La evaluación del desempeño directivo es parte de la evaluación de la gestión
4.2. Componentes de la evaluación del desempeño directivo
Unlike the scalars, in adding or subtracting vectors, you have to consider both sizes and directions of the vector quantities under consideration.
• Resultant vector
A single vector which has the same effect as two or more vectors acting in the same direction is called the resultant vector. Addition of vectors gives rise to a resultant vector.
Examples
Consider the following examples
i) Vectors acting in a straight line ii) Vectors acting in the same direction
Suppose a man walks 2 km due East and his wife walks 3 km due East.
What is the resultant vector (velocity)?
Solution
Let us represent the magnitude by the scale I cm = l km; and direction yan arrow head. Therefore, the man covering the distance of 2 km and his wife 3 km both due East would be represented thus:
The resultant vector (R) would take into consideration the magnitude and directions of both journeys as shown below:
10km/h 5 km/h
15km/h
10 km/h 5 km/h
his wife
2 cm B 3 cm
A Man
→ A
→ + B
5 cm
2cm 3 cm =
→ C
Fig. 1.3: Resultant vector (in one direction) b) Vectors acting in opposite directions
Suppose the man in our example above walks due East while his wife walks due West, the resultant vector would then be as shown in the diagram below:
Fig. 1.4: Resultant vector (in opposite direction)
3.4.3 Vectors Inclined at an Angle
Let us consider a man starting a journey from a point A and walks 3 km due North, then he turns and walks 5 km due East. What would be his Resultant Vector (displacement)?
• To find the resultant vector, you would either employ the Pythagoras rule (because the vectors are at right angle to each other) or by scale drawing.
Solution
(i) Using Pythagoras rule
Scale l cm: l km
Fig. 1.5: Resultant vector using the Pythagoras rule
• The magnitude of the resultant is given by:
→ → →
1 cm
= 1 km due west 3 cm
(3km)
→R 2 cm +
(2km)
→ A
→ B
N
A α
A E 2= A N2+NE2
∴ R = √ 32+52 = √ 9+25
= 34 = 5.83 km
• To obtain the direction (angle ) we use the relation opp 5
tan - = = 1.667 Adj 3
= tan -1 1.667 = 59.04o
• The result of this calculation implies that the resultant vector is 5.83km, North 59.04° East.
(ii) Using scale drawing
Here you should use a scale of say l cm to represent l km to find the resultant vector following the directives given below:
• draw 3 cm due North to represent 3 km due North;
• draw 5 cm due East from the head of 3 km due North to represent the 5 km;
• join the tail of 3 kmN to the head of 5 kmE;
• using your ruler, measure the distance AC and convert the result by the scale used; and find the direction by measuring the angle
∝ using your protractor.
R = 5.8 cm x l km = 5.8 km = 59°±1°
Fig. 1.6: Resultant vector using scale drawing
• Note that in general there are three methods of adding or compounding vectors inclined at angles to each other for the purpose of finding the resultant. These are:
α
(i) the parallelogram method;
(ii) the analytical method.
(iii) the triangle method.
(i) The Parallelogram method
In this method you will determine the resultant of two vectors inclined to each other at an angle from the diagonal of a parallelogram drawn with the two vectors as adjacent sides.
These two vectors must be drawn from a common origin.
The parallelogram law of vectors states that if two vectors are represented in magnitude and direction by the adjacent sides of a parallelogram, the diagonal of the parallelogram drawn from the point of intersection of the vectors represents the resultant vector in magnitude and direction.
Worked example
Find the resultant of two vectors 2N and 5N acting at a point 0 at an angle of 60° to each other.
Solution
• By Scale drawing
- using a scale (1 cm = l km or any other convenient scale) draw a horizontal line;
- using your ruler, measure (mark) out 5 cm to represent the 5N;
- using protractor, measure (mark) out angle 60° to the horizontal;
- draw a line to correspond with the 60° mark;
- use the ruler to measure (mark) out 2 cm on the line OP to represent the vector 2N;
- complete the parallelogram by drawing PC parallel and equal to OQ;
QC parallel and equal to AC;
- join OC ;
- use the ruler to measure the magnitude OC and the protractor to measure the direction (angle ∝).
• The length of OC gives the magnitude while angle ∝ gives the direction of the resultant vector.
• Your diagram should look like this.
P C
2N
(2cm) R R = 600 =
0 5N (5cm) Q
Fig. 1.7: Parallelogram method of determining R
• The Analytical method
You can also obtain the resultant vector by Cosine rule. In this case, you only have to draw a sketch of the parallelogram and use the Cosine rule and Sine rule to find the direction.
• Cosine Rule: R2 = OP2+ 0Q2 + 2 OP. OQ Cos 60
• Sine Rule: Sin ∝ = Sin B QC OC
• To obtain the magnitude using Cosine Rule
Fig. 1.8: The Cosine rule of obtaining the resultant OC 2 = OP 2 +OQ 2 + 2 OP x OQ x Cos60°
OC = OP 2 + OQ 2 + 20P x OQ Cos60°
= 22 + 52 + 2 x 2 x 5Cos60o 2N
0
C
β1200 R
600
P
5N Q
= 4 + 25 + 20 x 0.0.5
= 29 + 10
R = OC = 39 = 6.25N
• To obtain the direction using Sine rule Sin = Sin120° = 0.8660 QC OC OC
Sin 120° = Sin (180 - 120°) = Sin 60°
Sin = QC x 0.8660 OC
= 2 x 0.8660 = 1.732 = 0.2656 6.25 6.25
= Sin-1 0.2656
= 15.4°
(ii) The Triangle method
You can find the resultant of two vectors A and B inclined at an angle θ to each other by using the triangle method. To do this, you have to follow these steps:
• Starting from a point 0, draw OP (to scale) to represent the vector A
• Next, draw the second vector B i.e. PQ (to scale) by placing its tail at the head of the first vector A, ensuring that it is inclined to A by the angle θ, and
• Finally, the head of B is joined to the tail of A to make up the third side of the triangle OPQ. OQ represents the resultant of the two vectors A and B.
• The arrows on OP, PQ and QO follow one another round the triangle. The angle P0Q gives the direction of the resultant with respect to the vector A (the above is known as the triangle law of addition of vectors)
Worked example
• Two forces 3N and 2N are inclined to each other at 30°. Find the resultant force by the triangle method.
Solution
Let us use a scale of l cm = 1N
• first, we draw a horizontal line OA = 3 cm to represent the 3N force;
• from the tip (head) of A, we measure out angle 30°;
• next, we draw a line 2 cm to represent the 2N along the 30° mark (line of inclination to OA extended at 30° mark); and
• finally, we join point 0 to point B.
The expected result is as shown
Scale: 2 cm = 1N
Fig. 1.9: The triangle rule
Measure R and to obtain the magnitude and direction of the resultant respectively.
SELF ASSESSMENT EXERCISE 3
1. Outline the differences between Scalars and vectors
2. Draw vector diagrams to represent the following vector quantities (i) A vector of 5 km due South
(ii) A vector of I0 km on a bearing of 135°.
3. Find the resultant of the following vectors (i)
5N 7N
(ii)
3N 3N
(iii) 2N 5N
Fig. 1.10: Vectors in the same and opposite directions
4. A captain directs a ship due North at 100 km h-1, while the river is flowing due East at 30 km h-1. What is the resultant velocity of the ship?
5. Find the resultant of two vectors A and B of 3N and 4N acting at a point 0 at an angle of 450 to each other. Use:
(i) Parallelogram method (ii) Triangle method.