Vectors In The Plane Homework Meme

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Vectors in a plane


Definition of a vector in a plane

A  vector  in a plane is a  directed straight line segment  in the plane.

Any straight line segment in a plane is defined by the two its endpoints.  A straight line segment in a plane become the  directed  straight line segment when one of its two endpoints is declared as the  initial  point while the other endpoint is declared as the  terminal  point.

Vectors are named by their initial point and terminal point usually.  In naming vectors by their initial point and terminal point the initial point goes first.
Thus the  vector AB  in a plane is the directed straight line segment with the initial point  A  and the terminal point  B  (see  Figure 1).

Each ordered pair of points  (A,B)  in a plane defines the vector  AB  which is the directed segment  AB  with the initial point  A  and the terminal point  B.
A vector is drawn as a line segment with an arrow at the terminal point as shown in the  Figure 1.
Sometimes vectors are named by a single lover-case letter like  a  instead of  AB.

Each vector in a plane has the  length  (or the  magnitude)  and the  direction.  The length of the vector is the length of its straight line segment.  It is a non-negative real number.  The length of a vector  a  is usually denoted as  |a|.

    
    Figure 1.  The vector in a plane              



Figure 2.  Two equal vectors AB and CD                
                      in a plane


Figure 3.  Two opposite vectors AB and CD
                      in a plane

Two vectors  AB  and  CD  in a plane are called  equal  (or  equivalent),  AB=CD,  if the segments  AB  and  CD  are parallel and co-directed, and have equal length  (see Figure 2).
Two vectors in a plane are called  opposite  if the corresponding segments are parallel, have opposite directions, and have equal length  (see Figure 3).

It is clear that if the vector  AB  is equal to the vector  CD  and the vector  CD  is equal to the vector  EF  then the vector  AB  is equal to the vector  EF.

A vector whose initial and terminal points coincide is called the  zero vector.  The standard designation for the zero vector is  0.
The length of the zero vector is zero:  |0| = 0.  The direction of the zero vector is assumed to be arbitrary.

From the point of view of this definition of vectors equality,  the concrete location of the vector's initial and terminal points does not matter.  For example,  two directed straight line segments in the  Figure 2  represent the same vector.  Moreover,  there are infinitely many straight segments in a plane having the same length and direction.  All of them are distinguished by the location of the initial point only,  and all of them represent the same unique vector.  If you want, you may think that one of these vectors, which has the origin of the coordinate system as the initial point, represents all of this infinite set of equivalent vectors.  Only the direction and the magnitude do matter in distinguishing vectors.

For the quantitative definitions of the vector length and direction in a coordinate plane see the lesson  Vectors in a coordinate plane  under the current topic in this site.

Definition of adding vectors in a plane


If the endpoint of the vector  AB  is the initial point of the vector  BC  then the sum of vectors  AB  and  BC  is the vector  AC.  It is shown in  Figure 4.

    
  Figure 4.  The sum of the vectors          
  AB + BC = AC  (the  triangle rule)

    
  Figure 5.  The sum of the vectors          
AB + AD = AC  (the  parallelogram rule)

    
  Figure 6.  The sum of the vectors          
  AB + EF = AB + BC = AB + AD = AC

This is the  triangle rule  of adding vectors.   If the vectors  AB and  BC are two sides of the triangle  ABC,  then the sum
                                                    of the vectors  AB  and  BC  (in this order and orientation)  is the third side  AC  of the triangle
.

Now,  there is the  parallelogram rule  of adding vectors.   If two vectors  AB  and  AD  have the common initial point  A,  then the sum
                                                    of the vectors  AB  and  AD  is the vector  AC  which is the diagonal of the parallelogram  ADCB
                                                    built on the sides  AB  and  AD
.     It is shown in  Figure 5.

At last,  if there are two vectors  AB  and  EF  with no common initial points or endpoints,  as it is shown in the  Figure 6,  you can move/translate one of
the vectors,  let say the vector  EF,  parallel to itself in such a way to superpose its initial point with the initial point or with the endpoint of the vector
AB,  and then to apply the  parallelogram rule  or the  triangle rule,  whichever is applicable appropriately.

Due to properties of parallelograms the final result of vectors adding does not depend on which of the two rules you apply.
I mean these two properties of parallelograms:
    1)  if the quadrilateral has two opposite sides congruent and parallel,  then the quadrilateral is a parallelogram,   and
    2)  in a parallelogram, each diagonal divides it in two congruent triangles.

For the mentioned properties of parallelograms see the lessons
    Properties of the sides of a parallelogram  and
    In a parallelogram, each diagonal divides it in two congruent triangles
in this site.


The operation of adding vectors has remarkable properties.

1)  a + b = b + a  for any two vectors  a  and  b  in a plane  (commutativity property of addition).

    Indeed, translate the vectors  a  and  b  in a plane parallel to themselves to superpose their initial points and then apply the  parallelogram rule.

2)  (a + b) + c = a + (b + c)  for any three vectors  a,  b  and  c  in a plane  (associativity property of addition).

    For the proof, translate the vectors  a,  b  and  c  in a plane parallel to themselves to superpose the initial point of  b  with the endpoint of  a  and
    the initial point of  c  with the endpoint of  b  and then apply the  triangle rule. You will get that the sum of the vectors  a + b + c  is the vector with
    the initial point of  a  and the endpoint of  c  after the translations, and does not depend on placing parenthesis in the sum of the three vectors.

3)  a + 0 = a  (for any vector  a  adding the zero vector to  a  does not change the vector).

    It follows the definition immediately.

4)  a + (-a) = 0  for any vector  a  in a plane  (existence an opposite vector).

    It follows the definition immediately.

Definition of subtracting vectors in a plane


If the vectors  AB  and  AC  have the common initial point  A  then the difference of the vectors  AB  and  AC  is the vector  CB  which initial point is the endpoint of the vector  AC  and endpoint is the endpoint of the vector  AB.  It is shown in  Figure 7.

    
        Figure 7.  The difference           
      of the vectors AB - AC = CB

        
            Figure 8.  b + (a-b) = a


The operation of subtracting vectors has remarkable properties.

1)  b + (a-b) = a  for any two vectors  a  and b.

    For the proof,  use the  Figure 8.

Definition the product of a vector by a real number


The product of a vector  a  by a real number  r  is the vector  ra  whose length is equal to  |ra| = |r|*|a|  and whose direction coincides with that of the vector  a  if r >= 0  or is opposite to the direction of the vector  a  if r < 0.

If  r=0  or  a=0  then the product  ra  is the zero vector,  i.e.  its length is zero and the direction is undetermined.

The operation of product of a vector by a real number has the following properties.

1)  r(a + b) = ra + rb  for any vectors  a  and  b  and for any real number  r  (distributivity of product with respect to addition of vectors).

    You can prove this formula using the similarity properties of triangles.

2)  (r+s)a = ra + sa  for any vector  a  and for any real numbers  r  and  s  (distributivity of product with respect to addition of constants).

3)  r(sa) = (rs)a  for any vector  a  and for any real numbers  r  and  s.

4)  1a = a  for any vector  a  (multiplication by unity).


For examples of solved problems on summing vectors see the lessons
    - Sum of the vectors that are coherently oriented sides of a convex closed polygon,
    - Sum of the vectors that are coherently oriented sides of an unclosed polygon,  and
    - Sum of the vectors that connect the center of a parallelogram with its vertices
under the current topic in this site.


My introductory lessons on vectors in this site are
    - Vectors in a plane                                                                                                                        (this lesson)
    - Sum of vectors that are coherently oriented sides of a convex closed polygon
    - Sum of vectors that are coherently oriented sides of an unclosed polygon
    - Sum of vectors that connect the center of a parallelogram with its vertices
    - Vectors in a coordinate plane
    - Addition, Subtraction and Multiplication by a number of vectors in a coordinate plane
    - Summing vectors that are coherently oriented sides of a convex closed polygon
    - Summing vectors that are coherently oriented sides of an unclosed polygon
    - The Centroid of a triangle is the Intersection point of its medians
    - The Centroid of a parallelogram is the Intersection point of its diagonals
    - Sum of vectors connecting the center of mass of a triangle with its vertices
    - Sum of vectors connecting the center of mass of a quadrilateral with its vertices
    - Sum of vectors connecting the center of mass of a n-sided polygon with its vertices
    - Sum of vectors connecting the center of a regular n-sided polygon with its vertices
    - Solved problems on vectors in a plane
    - Solved problems on vectors in a coordinate plane
    - HOW TO find the length of the vector in a coordinate plane

Use this file/link  ALGEBRA-II - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-II.


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Дэвид Беккер стоял в центре пустого зала и думал, что делать. Весь вечер оказался сплошной комедией ошибок. В его ушах звучали слова Стратмора: Не звони, пока не добудешь кольцо.

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