Figure 1.1 Vector A and Vector F


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1 CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have only magnitude and completely specified by value and unit. Some examples for scalar quantities such as time, mass, speed, distance, work, power, energy, temperature, etc. For example time, time is specified by number and its unit, like 1 second, 1 hour, or 1 year. The operation of a scalar quantity is the same with algebraic operations such as addition, subtraction and multiplication. The second type of quantities is vector quantities, quantities which have magnitude and direction. Vector quantities are expressed by number, unit, and direction. Examples of vector quantities are displacement, velocity, acceleration, force, momentum, impulse, electric field, etc. For example, a car has velocity 70 km per hour eastward. The operations of vector quantities are different with scalar quantities. Vector addition and multiplication have their own rules, vector rules. A. Representation of Vector A vector quantity is symbolized with bold letters or regular letters marked with an arrow on it and the representation of a vector quantity is represented with an arrow. The direction of a vector is shown by direction of the arrow and its length of line represents the magnitude of the vector. For example, look at vector acceleration A and vector force F below. F A Figure 1.1 Vector A and Vector F Vector A has magnitude written as A or and its direction to the right while vector F has magnitude written as F or with direction upward of horizontal. In cartesian system of coordinates, vectors are shown as Figure 2.2. The vectors may be resolved into its component vectors. Vector S in cartesian coordinates two dimensions
2 may be resolved into two component vectors, and whereas in three dimensions, vector T may be resolved into three,, and. S T (a). Vector S in 2 dimensions (b). Vector T in 3 dimensions Figure 1.2 Vectors in cartesian coordinates. Vector S is formed by two component vectors and written as = + Where = component vector of S in x axis = component vector of S in y axis The magnitude of vector S is written as S and the magnitude of its component vectors are and. By using pythagorean theorem, it s found that = + Whereas vector T is formed by three component vectors and written as = + + where = component vector of T in x axis = component vector of T in y axis = component vector of T in z axis The magnitude of vector T, T may be stated in the form of its component vectors magnitude,, and. T is derived using Pythagorean Theorem = + +
3 B. Unit Vector A unit vector is a dimensionless vector having a magnitude of exactly one. The symbol for a unit vector is the same with common vector with a hat on it. In cartesian coordinates, it is defined unit vectors corresponding to each axis.,, and is a unit vector in positive x axis, positive y axis, and positive z axis, respectively, as shown in Figure 1.3. z y Figure 1.3 Unit vectors in cartesian coordinates Thus, by using unit vectors in cartesian coordinates, the unit vector notation for the vector S and vector T are = S " + S # = + + Generally, a unit vector of a particular vector is obtained by dividing the vector itself with its own magnitude. For example, a unit vector of T is symbolized with $ and obtained by dividing vector T with T $ = $ = C. Vector Addition Two vectors or more of the same kind can be added to form a resultant of vector. There are several methods to add vectors: parallelogram method, triangle method, and component method.
4 1. Parallelogram Method % ' ' (a) Figure 1.4 Addition of vector A and B with parallelogram method Vector A and B in Figure 1.4 (a) are the same vector so that can be added. To do parallelogram method, a parallelogram is made by lines A and B, and the diagonal line between vector A and B is the resultant vector R of A and B, (Figure b). % = + Since the angle between A and B is ', you can proof that the magnitude of R using cosines rule is ( = )* + +, + 2*, cos ' 2. Triangle and Polynomial Method (b) Two vectors and that is the same kind in Figure 1.5 (a) will be added. To add vector to vector using triangle method, first we draw vector and then draw vector with its tail starting from the tip of. The resultant vector R, where % = + is the vector drawn from the tail of to the tip of, as shown in Figure 1.5 (b). % % (a) (b) (c) Figure 1.5 Triangle method The order in which and are added is not significant, so that % = + = + (Figure c). Furthermore, to add more than two vectors, for example + + 1, polygon method is used to have resultant vector, % = + + 1, as shown in Figure 1.6.
5 1 % 1 Figure 1.6 Polygon method When three or more vectors are added, their sum is independent of the way in which the individual vectors are grouped together. This is called the associative law of addition + ( + 1) = ( + ) Component Method Now let us see how to use components to add vectors when the graphical method (parallelogram, triangle, and polygon methods) is not sufficiently accurate. Suppose we wish to add vector to vector. Both of them are vectors having two components in xy plane. Thus, the unit vector notation for the vector and are = * + * =, +, % = + % = 4* + * 5 + (, +, ) % = (* +, ) + (* +, ) % = (* +, ) + 4* +, 5 Since % in the unit vector notation is = ( + (, we can see that the components of the resultant vector are ( = * +, ( = * +, D. Vector Multiplication If vector is multiplied by a positive scalar quantity 6, then the product 6 is a vector that has the same direction as and magnitude 6*. If vector is multiplied by a negative scalar quantity 6, then the product 6 has opposite direction to and magnitude 6*. However, multiplying a vector to another vector is more complicated than multiplying a vector to a scalar. There are three kinds of vector multiplication, dot
6 product, cross product, and dyadic or tensor product. Each product has their own rules. In this case, we only study about the first two. 1. Dot Product Dot product of two vectors yields a scalar quantity. Dot product of vector and vector is defined as = *, 9:; ' where ' is an angle between the two vectors. It tells us that dot product of two perpendicular vectors is zero. Geometrical interpretation of dot product of vector and vector implies a scalar multiplication between magnitude of vector with the projection magnitude of vector on vector (Figure 1.7 a), or a scalar multiplication between the projection magnitude of vector on vector B with magnitude of vector (Figure 1.7 b). ' * E = * cos ', D =, cos ' ' Figure 1.7 Geometrical interpretation of dot product If vector and vector are expressed in unit vector notation, then the dot product is = 4* + * 5 (, +, ) = *, ( ) + *, ( ) + *, ( ) + *, ( ) = *, + *, Dot product can be used to find the angle between two vectors by using the equation cos ' = *, Some properties of the dot product : 1. = 2. = = = 1 ( since ' = 0 ) 3. = = = 0 ( since ' = 90 A ) 4. and perpendicular if = C and and are not zero
7 2. Cross Product Cross product of two vector quantities yields a vector quantity, and defined as = *, sin ' IJ and its magnitude = *, sin ' where ' is an angle between the two vectors and IJ is a unit vector perpendicular to the plane formed by vector and vector. The direction of IJ depends on the directions of vector and vector. There is a rule how to find direction of IJ, that is called right hand rule. For example, cross product of and, as shown in Figure 1.8. z y Figure 1.8 Cross product of and = 1 1 sin 90 A IJ = IJ If we rotate a screw from the tip of vector to the tip of vector, it will move upper in the same direction with vector. Since the magnitude of IJ = 1 then IJ = = Some properties of cross product : 1. =, =, = 2. =, =, = 3. = = = 0 ( since ' = 0 ) 4. = 5. Vector parallel to vector if = 0, and and are not zero Example Given vector K = and vector L = 2 2 +, a. draw vector K and L in cartesian coordinates b. find the magnitude of vector K and L c. find dot product of vector K and L d. find the product of K L e. find the angle between vector K and L f. find a unit vector perpendicular to the plane formed by vector K and L
8 Solution a. Vector K and L in cartesian coordinates L 2 2 z K 2 y b. Magnitude of vector K M = M + M + M M = M = 9 M = 3 b. Magnitude of vector L O = O + O + O M = )2 + ( 2) + 1 M = 9 M = 3 c. Dot product of vector K and L K L = K L = K L = 2 d. The product of K L K L = K L = 2 + ( 2 ) ( 2 ) + 2 K L = ( 2 ) + 2 K L = C C P + P + C K L = Q + R 6 e. The angle between the two vectors is K L sin ' = MO sin ' = ) ( 6) 3 3 sin ' = 81 9 sin ' = 1 then, ' = 90 A f. Suppose the unit vector is IJ, then IJ = K L K L
9 IJ = U IJ =
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