2- Scalars and Vectors
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1 2- Scalars and Vectors Scalars : have magnitude only : Length, time, mass, speed and volume is example of scalar. v Vectors : have magnitude and direction. v The magnitude of is written v v Position, displacement, velocity, acceleration and force are examples of vector quantities. is example of vector quantities. v 1
2 Properties of Vectors Equality of Two Vectors Two vectors are equal if they have the same magnitude and the same direction Movement of vectors in a diagram Any vector can be moved parallel to itself without being affected 2
3 Negative Vectors Two vectors are negative if they have the same magnitude but are 180 apart (opposite directions). Multiplication or division of a vector by a scalar results in a vector for which (a) only the magnitude changes if the scalar is positive. (b) the magnitude changes and the direction is reversed if the scalar is negative. 3
4 Components of a Vector These are the projections of the vector along the x- and y-axes A A A x y 4
5 The x-component of a vector is the projection along the x-axis The y-component of a vector is the projection along the y-axis Then, A Acos x A Asin y A A A x y A A 2 x A 2 y and tan 1 A A y x 5
6 Unit Vectors A Unit Vector is a vector having length 1 and no units It is used to specify a direction. Unit vector u points in the direction of U Often denoted with a hat : u = û û U = U û Useful examples are the cartesian unit vectors [ i, j, k ] Point in the direction of the x, y and z axes. R = r x i + r y j + r z k 6
7 Adding Vectors When adding vectors, their directions must be taken into account and units must be the same First : Algebraic Methods Second : Graphical Methods 7
8 Adding Vectors Algebraically (1)Choose a coordinate system and sketch the vectors (2)Find the x- and y-components of all the vector (3)Add all the x-components This gives R x : R x v x (4)Add all the y-components This gives Ry R y v y R x R y (5)find the magnitude of the Resultant 2 2 R R Use the inverse tangent function to find the direction of R: tan 1 R R y x 8
9 Adding Vectors Graphically (Triangle Method) Continue drawing the vectors tip-to-tail The resultant is drawn from the origin of A to the end of the last vector Measure the length of R and its angle 9
10 When you have many vectors, just keep repeating the process until all are included The resultant is still drawn from the origin of the first vector to the end of the last vector 10
11 When you have only two vectors, you may use the Parallelogram Method All vectors, including the resultant, are drawn from a common origin Alternative Graphical Method (Parallelogram Method) The remaining sides of the parallelogram are sketched to determine the diagonal, R 11
12 Vector Subtraction Special case of vector addition If A B, then use A+(-B) Continue with standard vector addition procedure 12
13 Example : Vector A has a magnitude of 8.00 units and makes an angle of 45.0 with the positive x axis. Vector B also has a magnitude of 8.00 units and is directed along the negative x axis. Using graphical methods, Find: (a) the vector sum A + B. (b) the vector difference A - B. Solution: 13
14 Example : Given the vectors : A = 2.00 i j and B = 3.00 i j, (a) draw the vector sum, C = A + B and the vector difference D = A - B. (b) Calculate C and D, first in terms of unit vectors and then in terms of polar coordinates, with angles measured with respect to the, +x axis. Solution: C A B 4 tan i 4j D A ( B) i 8j tan
15 Example : A particle undergoes three displacements given by : d 1 (i 3j k)cm, d 2 (2i j 3k) cm d 3 ( i j) cm Find the resultant displacement of the particle Solution: R d1 d 2 d3 (1 R (2i 3j 4k)cm 2 1)i (3 11)j ( 1 3 0)k The resultant displacement has component Rx 2cm, Ry 3cm, Rz 4cm The magnitude is R R x R y R z 5. 39cm 15
16 Example : Consider the two vectors A = 3 i - 2 j and B = i - 4 j. Calculate: (a) A + B, (b) A - B, (c) A + B, (d) A - B, and (e) the directions of A + B and A - B. 16
17 Example : Three displacement vectors of a small ball are shown in Figure, where A = 20.0 units, B = 40.0 units, C = 30.0 units. Find the magnitude and direction of the resultant displacement Solution: d x 0 40cos 45 30cos unit d y 20 40sin 45 30sin 45 27unit R (d x ) 2 (d y ) unit tan d d y x
18 Example : Find the magnitude and the direction of resultant force 18
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