IB Math SL1 - Santowski T5.1 Geometric Vectors 3/3/2016 1 IB Math SL1 - Santowski.

18
IB Math SL1 - Santowski T5.1 – Geometric Vectors 06/17/22 1 IB Math SL1 - Santowski

description

(B) Visual Representation Let’s consider the two points: A(-3,2) which will be the starting point (or tail) of the vector and the point B(1,-1) which will be the ending point (or head) of the vector. So, we have constructed the vector The vector 3/3/ IB Math SL1 - Santowski

Transcript of IB Math SL1 - Santowski T5.1 Geometric Vectors 3/3/2016 1 IB Math SL1 - Santowski.

Page 1: IB Math SL1 - Santowski T5.1  Geometric Vectors 3/3/2016 1 IB Math SL1 - Santowski.

IB Math SL1 - Santowski

T5.1 – Geometric Vectors

05/08/231 IB Math SL1 - Santowski

Page 2: IB Math SL1 - Santowski T5.1  Geometric Vectors 3/3/2016 1 IB Math SL1 - Santowski.

(A) IntroductionWe have considered vectors as directed line

segments and investigated vectors:

(a) visually as scale drawings with a particular angle/direction associated with them

(b) algebraically in terms of x- and y-components as well as i- and j- components

(c) Now let’s place our vectors into the Cartesian plane and combine a visual representation with an algebraic representation as we introduce ordered pairs with which to work with vectors

05/08/232 IB Math SL1 - Santowski

Page 3: IB Math SL1 - Santowski T5.1  Geometric Vectors 3/3/2016 1 IB Math SL1 - Santowski.

(B) Visual RepresentationLet’s consider the two

points: A(-3,2) which will be the starting point (or tail) of the vector and the point B(1,-1) which will be the ending point (or head) of the vector.

So, we have constructed the vector

The vector

ABv

ABv

05/08/233 IB Math SL1 - Santowski

Page 4: IB Math SL1 - Santowski T5.1  Geometric Vectors 3/3/2016 1 IB Math SL1 - Santowski.

(B) Visual RepresentationLet’s consider the two

points: A(-3,2) which will be the starting point (or tail) of the vector and the point B(1,-1) which will be the ending point (or head) of the vector.

So, we have constructed the vector

The vector

ABv

ABv

05/08/234 IB Math SL1 - Santowski

Page 5: IB Math SL1 - Santowski T5.1  Geometric Vectors 3/3/2016 1 IB Math SL1 - Santowski.

(C) Working with Geometric VectorsSince is defined on the Cartesian

plane, we can determine:

(a) its components

(b) its length

(c) its direction

ABv

05/08/235 IB Math SL1 - Santowski

Page 6: IB Math SL1 - Santowski T5.1  Geometric Vectors 3/3/2016 1 IB Math SL1 - Santowski.

(C) Working with Geometric VectorsSince is defined on the Cartesian

plane, we can determine:

(a) its components (+4 in the x and -3 in the y)

(b) its length (c) its direction

ABv

22 34

127 bearing 3743tan 1

05/08/236 IB Math SL1 - Santowski

Page 7: IB Math SL1 - Santowski T5.1  Geometric Vectors 3/3/2016 1 IB Math SL1 - Santowski.

(D) Working with Geometric Vectors – Geometric ShapesGiven the following diagram, use vector

methods prove that the figure is a parallelogram

05/08/237 IB Math SL1 - Santowski

Page 8: IB Math SL1 - Santowski T5.1  Geometric Vectors 3/3/2016 1 IB Math SL1 - Santowski.

(D) Working with Geometric Vectors – Geometric ShapesGiven the following diagram, use vector

methods to prove that the figure is a parallelogram

HINT: What does vector equality mean? How can you show 2 vectors are equal?

05/08/238 IB Math SL1 - Santowski

Page 9: IB Math SL1 - Santowski T5.1  Geometric Vectors 3/3/2016 1 IB Math SL1 - Santowski.

(D) Working with Geometric Vectors – Geometric Shapes

Given the points A(-3,2), B(2,1) and C(-1,-4), find the position of the fourth point such that ABCD is a parallelogram

05/08/239 IB Math SL1 - Santowski

Page 10: IB Math SL1 - Santowski T5.1  Geometric Vectors 3/3/2016 1 IB Math SL1 - Santowski.

(D) Working with Geometric Vectors – Geometric ShapesGiven the following diagram, use vector

methods prove that the figure is a parallelogram

05/08/2310 IB Math SL1 - Santowski

Page 11: IB Math SL1 - Santowski T5.1  Geometric Vectors 3/3/2016 1 IB Math SL1 - Santowski.

(D) Working with Geometric Vectors – Geometric ShapesGiven the following diagram, use vector

methods determine the position of the 4th point so that the figure is a parallelogram

05/08/2311 IB Math SL1 - Santowski

Page 12: IB Math SL1 - Santowski T5.1  Geometric Vectors 3/3/2016 1 IB Math SL1 - Santowski.

(E) Position VectorsIn introducing the idea

of a position vector, we now change from free vectors (the vectors position in space is NOT considered) to fixed vectors (which start at a SPECIFIC point and are thus fixed in space)

The most convenient fixed point the origin

05/08/2312 IB Math SL1 - Santowski

Page 13: IB Math SL1 - Santowski T5.1  Geometric Vectors 3/3/2016 1 IB Math SL1 - Santowski.

(E) Position VectorsIn introducing the idea

of a position vector, we now change from free vectors (the vectors position in space is NOT considered) to fixed vectors (which start at a SPECIFIC point and are thus fixed in space)

Consider point P and construct the position vector

OP

05/08/2313 IB Math SL1 - Santowski

Page 14: IB Math SL1 - Santowski T5.1  Geometric Vectors 3/3/2016 1 IB Math SL1 - Santowski.

(E) Position VectorsNow construct the 2

position vectors

So the three vectors can be connected via algebraic operations:

and our vector can be viewed as a result of a vector subtraction

ON

OM

NONMMO

MONONM

05/08/2314 IB Math SL1 - Santowski

Page 15: IB Math SL1 - Santowski T5.1  Geometric Vectors 3/3/2016 1 IB Math SL1 - Santowski.

(F) Collinear PointsWe can work with

position vectors to prove that three points are collinear (in this case A, R, B)

Let’s work with :(i) A(2,1), R(4,7), &

B(12,16)

(ii) A(2,1), R(4,7), & B(12,12)

05/08/2315 IB Math SL1 - Santowski

Page 16: IB Math SL1 - Santowski T5.1  Geometric Vectors 3/3/2016 1 IB Math SL1 - Santowski.

(F) Collinear PointsWe can work with position vectors to

prove that three points are collinear (in this case A, R, B)

Let’s work with : (i) A(2,1), R(4,7), & B(12,16)

(ii) A(2,1), R(4,7), & B(12,12)

Conclusion to be made 3 points are collinear if one position vector can be written as sum of the other 2 position vectors

i.e. r = as + tb

05/08/2316 IB Math SL1 - Santowski

Page 17: IB Math SL1 - Santowski T5.1  Geometric Vectors 3/3/2016 1 IB Math SL1 - Santowski.

(G) 3D SpaceCo-ordinate geometry can also be used to

introduce 3D space as we “extend” our Cartesian plane into a third dimension as we consider our vector (ax, ay, az)

05/08/2317 IB Math SL1 - Santowski

Page 18: IB Math SL1 - Santowski T5.1  Geometric Vectors 3/3/2016 1 IB Math SL1 - Santowski.

HomeworkHW – Ex 15C.1 #1bc, 2ade; Ex 15C.2 #1f, 2ef, 3; Ex 15C.3 #1df; Ex 15C.4 #1ace, 2fgh;

Ex 15E #1,2, 3, 4, 5ab, 6,7

05/08/2318 IB Math SL1 - Santowski