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Vector quantities consist of both magnitude and direction. The magnitude is made up of a number and a unit. Scalar quantities consist of only a magnitude. Position, displacement, velocity, acceleration, force, weight, and momentum are examples of vector quantities. Scalar quantities include such things as mass, time, distance, speed, work, and energy, to name but a few. Vector quantities are represented on a diagram by a directed line segment, drawn to scale, with reference coordinates to show direction. The tail of a vector is called the origin and the tip is called the terminal point. Equivalent vectors have the same magnitude and direction. Collinear vectors can be added algebraically or graphically. The resultant vector is obtained from vector addition. Graphical addition of vectors is performed on a neat, accurate, scale diagram. Non-collinear vectors exist in more than one dimension. The sum of any two or more vectors can be determined graphically or mathematically. Vectors are added by aligning the tail of one vector (origin) with the tip of another vector (terminal point) on a neat, accurate, properly scaled diagram. The vector sum of two or more vectors is called the resultant vector. The resultant vector points from the tail of the first vector to the tip of the last vector being added. To make a vector negative, change its direction. A negative vector has the opposite direction of a positive vector. The magnitude and direction must be stated for the resultant vector, as for any other vector quantity. Vector and scalar operations yield very different results. The two must not be confused.
To add vectors there are two techniques available, geometric addition and algebraic addition. Both yield the same result. The choice of which technique to use in adding vectors depends on the application and is a matter of convenience. Since a vector is defined by its magnitude and direction, changing its location in our reference frame without changing its direction or magnitude leaves it the same vector. We are free to relocate a vector anywhere in our space where we find it convenient. To add vectors geometrically you just place the tail of one at the head of the other. The sum then is a vector from the tail of the first vector to the head of the last.
The algebraic addition of vectors involves simply adding up the like components of the vectors. Imagine a vector with its tail at the origin. The scalar components of that vector are just the coordinates of the head of the vector. Remember that the coordinates where the distances along each axis which defined the position of the head of the vector. To add two vectors, add all the x components together and all the y components. The algebraic addition of vectors works because the sums of the components are the components of the sum.
To break a vector down into its components is called vector resolution. What that means is that it will resolve a vector into its ¡§vector components¡¨ which as stated before the coordinates of the head of the vector when its tail is at the origin. The vector components are the vectors lying along the axes, which add up to the vector that you interested in. Since a vector along an axis has its direction fixed by definition, only its magnitude changes. That means you can uniquely define each component of a vector with just a number, knowing whether we are talking about the x, y or z component. Remember that you could also specify the position of a particle with the scalars, which are called coordinates. There is an obvious relationship between the coordinates of a position and the components of the vector pointing to that position. They are numerically equal. Write your Scalars and VectorsReport research paper
Vectors can be related to the basic coordinate systems, which are used by the introduction of what is called unit vectors. A unit vector is one, which has a magnitude of one and is often indicated by putting a hat on top of the vector symbol. You can define a unit vector in the x-direction by X hat or it is sometimes denoted by I hat. Similarly in the y-direction, you use Y hat or sometimes j hat. Employing multiples of the unit vectors, X hat and Y hat, can now represent any two-dimensional vector. The vector A can be represented algebraically by A = Ax + Ay. Where Ax and Ay are vectors in the x and y directions. If Ax and Ay are the magnitudes of Ax and Ay, then Ax x hat and Ay y hat are the vector components of A in the x and y directions respectively. The breaking up of a vector into its component parts is known as resolving a vector. The representation of A by its components, Ax x hat and Ax y hat is not unique. Depending on the orientation of the coordinate system with respect to the vector in question, it is possible to have more than one set of components.
The scalar product of two vectors a and b is defined to be the real number a & b given by |a||b cos á whereánis the angle between a and b. This is called the scalar product since you can take vectors and produces a scalar (a real number). It is also called the scalar product to distinguish it from the vector product. Since cosá = cos (-á) it is not important to worry about about whether the angle is positive or negative. The relationship between a and b does not need to have the same units. We also need to notice that AB=BA. Which means that A(B+C)=AB+AC.
Given any two vectors A and B, the vector product A x B is defined as a third vector. This is called the vector product or cross product. The magnitude of the vector product is defined as |AxB|=ABsin á. The direction of C=AxB is defined as being given by the right hand rule, whereby you hold the thumb, fore-finger and middle finger of your right hand all at right angles to each other. The thumb represents vector C, the forefinger represents A and the middle finger represents B. The order in which two vectors are multiplied in a cross product is important. AxB=-BxA. If you change the order of the vectors in a cross product fashion, you must change the sign. If A is parallel to B (á=0 degrees or 180 degrees), then AxB=0. If A is perpendicular to B, then |A x B|=AB. The vector product obeys Ax(B+C)=
AxB+AxC.
Below are some examples of the previous material to make sense of it all.
1. Adding Vectors
A plane travels 50km due north and then 70km in a direction of 50 degrees west of north. Find the magnitude and direction of the planes displacement.
180degrees-50 degrees=10degrees
R=sqr(A^+B^-ABcosnáw
R=sqr(50^+5^-(505)cos10)
R=77.km
Sin/B=sin/R
á=(70/77.)sin10
á=.6
sin^-1(.6)=44degrees
The resultant displacement of the plane is 77.km in a direction of 44 degrees west of north.
. Components of a Vector
1. A particle undergoes three consecutive displacements
d1=(0i+40j+10k)cm, d=(5i-15j-4k)cm, and d=(-15i+17j)
Find the components of the resultant displacement and magnitude.
R=d1+d+d
R=(0+5-15)icm +(40-15+17)jcm +(10-4)kcm
R= (0i +4j + 6k)cm
The resultant displacement has components Rx=0cm,Ry=4cm,and Rz=6cm
R=sqr(Rx^+Ry^+Rz^)
R=sqr(0^+4^+6^)=5cm
Its magnitude is 40cm
. Scalar Product
1. The vectors A and B are given by A=4i+5j and B=-i+j. Determine the scalar vector AB
AB=(4i+5j)(-I+j)
=-4iI+4ij-5jI+5jj
=-4(1)+1(0)-5(0)+15(1)
=-4+15=11
4. Vector Product
1. Two vectors lying in the xy plane are given by the equations. Find the product and verify that A x B= -B x A
A=i+4j and B=-I+j
A x B = (i+4j) x (I +j)
= i x j + 4j x (-i) = k+4k=1k
B x A = (-i +j) x (i +4j)
= -I x 4j +j x i = -4k ¡V k= -1k
Therefore A x B = -B x A
There are two basic techniques when adding vectors geometric and algebraic. They will both give you the same result. To algebraically add vectors you just add up the like components. Vector A is represented by A=Ax + Ay where Ax and Ay are vectors in the x,y direction. The scalar product of two vectors A and B is defined by the relationship AB=AB cos Ö where the result is a scalar quanity and Önis the angle between the two vectors. The scalar product obeys the communitive and distributive laws. Given two vectors A and B, the cross product A x B is a vector C having a magnitude C=AB sin Ö where Ö is the angle between A and B. The direction of vector C=A x B is perpendicular to the plane formed by A and B and this direction is determined by the right hand rule.
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