Gr11 Lesson 2. Vectors and parallelogram. The parallelogram and its properties. 1. Definition: A parallelogram ABCD is a quadrilateral such that the segment ...
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MATHEMATICS Gr11 Lesson 2.
Vectors and parallelogram
The parallelogram and its properties. 1.
De…nition: A parallelogram ABCD is a quadrilateral such that the segment [AC] and [BD] have the same midpoint O: O is then called the centre of the parallegram ABCD
Remark: The order of the characters ABCD is very important ! For example, if ABCD is a parallelogram, ACBD is not a parallelogram.
We say that ABCD is a ‡at parallelogram when is a parallelogram and A, B, C and D are on the same line.
Exercise: Let ABCD and AECF be two parallelogram, prove that BEDF is a parallelogram. 2.
Elementary properties of the parallelogram. Most of the properties of the parallelogram come from its property of symmetry to its centre. (a) The opposites sides of a parallelogram have the same length. (b) The opposites sides of a parallelogram are parallel. (c) A quadrilateral ABCD is a parallelogram if and only if (AB) k (CD) and (AD) k (BC)
Vectors 1.
De…nition: Scalar quantities are quantities that have only magnitudes such as time, area and distance. Vector quantities are quantities that have both magnitudes and directions such velocity (speed and direction), force and acceleration. The directed line supporting the vector is called direction of the vector and the magnitude is called modulus of it. ! The vector 0 is the vector with length 0. It has no direction. A vector can be represented by a directed line segment, whose direction is given by the arrow and the length shows the magnitude of the vector. We denote a vector by adding an arrow on its name as ! u: Remark : We can de…ne vector using points as follows : ! For A and B two given points, the vector AB is the vector with Its direction is the line (AB) directed by the sense : A to B Its modulus the distance AB
2.
Equality of two vectors: (a) De…nition. Two vectors are equals if their magnitude are equals, their directions are parallel with the same sense.
(b) Case of vectors de…ned with points: If the vectors are de…ned with points: ! ! AB = CD if and only of ABDC is a parallelogram (c) Vector and midpoint.
! ! For three points A, B and I, I is the midpoint of [AB] if and only if AI = IB It comes from I is the midpoint of [AB] if and only if AIBI is a parallelogram and ! ! AI = IB if and only if AIBI parallelogram.
3.
Representation of a vector. Let O be a given point of the plan. For all vector ! u ; there exists one and only one point M ! ! such that u = OM
Examples: ! If ! u = 0 then M = O ! ! If ! u = AB, ! u = OM which is equivalent to OABM parallelogram. So we can represent the vector plan. 4.
Sum of two vectors. (a) De…nition. We de…ne an operation named addition on the set of vectors as follows : For three given points A, B and C of the plan. ! ! ! AB + BC = AC ( Chasles’s relation)
! ! ! AC is called the resultant of AB and BC (b) Algebric properties. For all vectors ! u ,! v and ! w ! ! ! u + 0 = u ! u +! v =! v +! u ! ! ! (u + v)+ w =! u + (! v +! w)
! For all vector ! u there is one and only one vector ! v verifying ! u +! v = 0 , this is the vector with the same modulus and a direction parallel to the direction of ! u but directed to the other sense. We name it the opposite of ! u and write it ! u: ! ! ! ! Remark: For all points A and B; AA = 0 and BA = AB The di¤erence of two vectors ! u and ! v , denoted ! u ! v equals ! u +( ! v ) and it results that for all points A, B and C ! ! ! BC = AC AB
(c) The rule of parallelogram Let O be a point of the plan and ! u and ! v two vectors, considering the points A and B ! ! ! de…ned ! u = OA and ! v = OB. The point M de…ned by OM = ! u +! v is such that AOBM is a parallegram. ! ! ! ! ! ! ! ! ! ! ! Proof: OM = OA + OB () OM OA = OB and OM OA = OM + AO = AM and ! ! then AM = OB which is equivalent to AOBM parallelogram.
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