Vectors Illustration

8/8/2019 Vectors Illustration

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Problems Illustrating IB Math ConceptsProblems Illustrating IB Math ConceptsProblems Illustrating IB Math ConceptsProblems Illustrating IB Math Concepts

TOPIC 5 - Vectors

5.1 Vectors as Displacements in the Plane5.1.1

Vectors (introduction)1) If A = ( 1, 2) and B = (3,4), find: (a) AB (b) AB (c) AB

2) If A = (-5,-2) and B = (-3,6), find AB

Answers: 1) (a) 4,2 (b) 4.472 (c) 27

2) 63

5.1.2

Vectors (introduction) basic operations, unit basis vectors, unit vector

1) If 2 3v i j= +

and 3u i j=

, find (a) 2u v

(b) 3u v+

2) Find the unit vector in the same direction as 3 2v i j=

Answers: 1) (a) 4 5i j

(b) 9 8i j+

2)3 2

13 13i j

5.1.3

Vector Addition, algebraic and graphical If 3, 2u = and 4,8v = , find 2 3u v+

.

Answer: 6,20

5.2 The Scalar (dot) Product of Two Vectors5.2.1

Dot ProductIf 3, 6u = , 4,2v = , and 12, 6w = , find 1) v u

i 2) v w

i

Answers: 1) 0 2) -60


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5.2.2

The scalar (dot) product of two vectors; coswvwv = ; 332211 wvwvwvwv ++=

perpendicular vectors, parallel vectors; the angle between two vectors; For non-zero perpendicular vectors 0= wv ; for non-zero parallel vectors wvwv =

1) Given

=

=

1

6

2

612

xwv ,find x such that the vectors are parallel.

Answer: Parallel vectors means one vector is a scalar multiple of the other. Thus, 2== k vwk .Thus, 3= x .

2) Given

=

=1

6

26

12

xwv , find x such that the vectors are perpendicular.

Answer: Perpendicular vectors have a scalar (dot) product of zero. Thus,

337

026)6(12 ==+ x x .

3) For x = 3, find the angle between v and w .

Answer: The angle between vectors is represented bywv

wv = cos . So,

8464

56

46184

56cos == .

Now 5.528464

56cos 1 =

= .

4) Given that the angle between two vectors is 30 , and their magnitudes are 5 and 6,find the exact value of the scalar product of the vectors.

Answer: The scalar product can be represented by coswvwv = . Thus,

31530cos65==

wv .

5) Point A is 20m north of the origin. A point P travels on a line parallel to ji 2+ .

(a) Write down the vector which represents the displacement from O to A.(b) Write down the vector that represents the displacement from O to P when P is x km east of

point O.

(c) Express AP in terms of x.(d) Hence, find the value of x when P is closest to A.


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Answers: (a)

=

20

0OA

(b) OP is parallel to

3

1. Thus,

=

x

xOP

3.

(c) According to vector geometry, OP APOA =+

==203 x

xOAOP AP .

(d) For P to be closest to A, then 0= OP APOP AP .

Thus, ( ) 0610060903203

22 ==+=

x x x x x x

x

x

x

60 == xor x (but we discard 0= x as it is not a reasonable answer).

5.3-5.4 Representation of a line as tbar += ; Lines in the plane and in three-dimensional space; Examples of applications: interpretation of t as time and b as velocity, with

b representing speed; The angle between two line; Distinguishing between

coincident and parallel lines; Finding points where lines intersect; Awareness that non-parallel lines may not intersect .

A spider and a fly crawl so that their positions at time t (in seconds) are given by:

spider: x

y

=3

4

+ t 2

3

fly:

+

=

1

2

6

9t

y

x

(a) Write down the velocity vector of the spider and find its speed.(b) Find the Cartesian equation of the path of the spider in the form ax + by + c = 0(c) Find the coordinates of the intersection of the paths.(d) At what time, if any, will the spider catch the fly?(e) Find the acute angle between the paths of the spider and the fly.

Answers: The equation of a line can be represented by vectors in the form tbar += . The vector a represents an initial position or point on the line, and could be any point on the line. The vectorb is related to the slope of the line and hence will represent not only its direction, but itsvelocity as well. Thus, the magnitude of the b vector will be its speed. Adding the two vectorsa and b will find a next point on the line, just like counting rise over run in basic

graphing. By using t as a multiplier of the slope vector, any point on the line could be found,depending on the value of t . Thus, tbar += represents an entire line and all its infinitepoints.

(a)

=

3

2v and speed 1332 22 =+= .


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(b)23

23

==

xt t x . By substituting into t y 34 += , we obtain

+=

23

34x

y .

Multiplying both sides by 2 (to eliminate the fractions), we obtain 9382 += x y which

then becomes 0123 =+ y x .

(c) In a similar manner as part (b) we find the Cartesian equation f the fly to be

032 =+ y x . Solving the system of equations123

32=+

=+

y x

y x, we obtain the coordinates of

( )2,1 .

(d) Just because the paths cross, does not mean the spider and the fly are at the same placeat the same time. We set the x-equations equal and the y-equations equal to find the times atwhich they are at the same coordinate. If both times are equal, they are at the same place at thesame time, and the spider catches the fly. So set 32923 =+= t t t , and

5.2634==+

t t t . Thus, the spider does not catch the fly.

(e) The angle between the paths is the angle between the direction vectors of their paths.Thus we use

15065

7cos

65

7

513

1

2

3

2

cos 1 =

=

=

= . The acute angle is 30 .

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Vectors Illustration