vector std

8/13/2019 Vector Std

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VECTORS

CJHL 2009 Page 1

2. VECTORS

2.1 Introductions

Scalar:Physical quantities which have only magnitude are called scalar

quantities. For example, speed, volume, mass, work done, energy,

temperature etc.

Vector:Physical quantities which possess both magnitude and direction are

called vector quantities. For example, displacement velocity,

acceleration and force etc.

Geomatrical representation of vectors:

A vector is customarily re[resented by a directed line segment from a point A

called initial point to another point B called terminal point.

We usually denote vectors by bold faced letter or letters with arrow over them.

For example;

A,

A ,

AB

The magnitude or modulus or length of the vector

A is denoted by

A ,

A ,

AB

Equal Vectors:

Two vectors

A and

B ara said to be equal I f they have the same magnitude

and the same direction regardless of their initial point

Negative vecto

A vector having direction opposite to that of a vector

A is denoted -

A or

PQ = -

QP

A

B

B

A

A

A

P

Q

P

Q


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Zero Vector:

If the initial and terminal points of a vector coincide, then the vector has

length zero, denoted by O

2.2 Vector in Space

The position vector

a with terminal point at P(a1, a2, a3) denoted by < a1, a2,

a3> as shown in figure 2.1

The vector

a =

OP = < a1, a2, a3> is the position vectorof the point

P(a1,a2,a3).

The vector with initial point at P( a1, a2, a3) and terminal point Q ( b1, b2, b3)

correspond to position vector

PQ =

The magnitudeof the position vector

a = < a1, a2, a3> is the length of its

representation and denoted by

2

3

2

2

2

1 aaaa ++=

Vector Addition

The addition of two vector

a = < a1, a2, a3> and

b = < b1, b2, b3> is

a +

b = < a1+ b1, a2+ b2, a3+ b3>

Graphical representation of vector addition are as shown below

Figure 2.1

b

a

a+bb

a

a+b

Parallelogram LawTriangle Law

O

x

y

z

P(a1, a2, a3)=

a


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Vector Difference

The difference of two vectors

a = < a1, a2, a3> and

b = < b1, b2, b3> is

a -

b = < a1- b1, a2- b2, a3- b3>

Graphical representation of vector substraction is

Scalar Multiplication:

If c is a scalar and

a = < a1, a2, a3> then the vector

c

a = c < a1, a2, a3> = < ca1, ca2, ca3>

The two vectors

a and

b are parallel if

b = c

a for some scalar c

Example 1

Find the vector represented by the line segment with initial point A(2, -3, 4)

and terminal point B(-2, 1, 1)

Example 2

If

a = and

b = , find a and the vector

a +

b ,

a

b ,

3

b , 2

a + 5

b

Example 3

Determine whether the given pair of vectors is parallel:

(a)

a = < 2,3,4> and

b = < 4,6,5>

(b)

a = < 2,3,4> and

b = < -4,-6,-8>

aa-b

-b


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Since V3is three-dimensional, these standard basis vectors are unit vector,

since

i =

j =

k= 1. Thus for any

a = < a1, a2, a3>, we can write

a = < a1, a2, a3> = a1

i +a2

j +a3

k

For example =

i 2

j + 6

k

Example 4

If

a =

i + 2

j - 3

k and b = 4

i + 7

k, express 2

a + 3

b in terms of

i ,

j and

k. Find

+ b3a2

1

1

1

i

j

k

x

y

z


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VECTORS

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Unit Vector:

Unit vector is a vector having 1 unit length. If

a is a any vector, then

=

a

a

u

is a unit vector in the direction of

a

Example 5

Find the unit vector in the direction of the vector 2

i

j - 2

k

Example 6

Find a vector with the given magnitude in the same direction as the given

vector

(a) magnitude 3,

v =

i + 2

j - 3

k

(b) magnitude 4,

v = 2

i

j - 2

k

Dot Product

Definition: The dot product of two vectors

a = and

b = in V3is defined by

a .

b = . = a1b1+ a2b2+ a3b3

Notice that the dot product of two vectors is a scalar


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Example 5

Compute the dot product

a .

b for

a)

a = and

b =

b)

a = and

b =

c)

a = i+ 2j 3k and

b = 2j k

The dot product in V3satisfies the following properties

If

a ,

b and

c are vectors in V3and c is a scalar, then

i)

a .

a = 2a

ii)

a . (

b +c) =

a .

b +

a . c

iii) 0 .

a =0

iv)

a .

b =

b .

a

v) (c

a ) .

b = c (

a .

b ) =

a .(c

b )

Theorem : If is the angle between the vectors

a and

b , then

a .

b = cosba

Example 6

Find the angle between the vectors

a)

a = and

b =

b)

a = and

b =

Two non zero vectors

a and

b are called perpendicular or orthogonal if the angle

between them is2

= . Then Theorem ?? gives

a .

b = 2cosba

= 0

Thus two vectors

a and

b are orthogonal if and only if

a .

b = 0

Example 7

Show that 2i+ 2j k is perpendicular to 5i 4j+ 2k


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Example 8

Determine whether or not the following pair of vectors are orthogonal.

a)

a = and

b =

b)

a = and

b =

Example 9

Find the angle between the vectors

a = and

b =

Example 10

Find the angle between the vectors u=i-2j+ 2k and

a) v= -3i + 6j+ 2k b) w= 2i+ 7j+ 6k

c) z= -3i+ 6j 6k

Cross Preoduct

Theorem: For two vectors

a = and

b = in V3,

we define the cross product of

a and

b to be

a x

b =

321

321

bbb

aaa

kji

=

32

32

bb

aai-

31

31

bb

aaj+

21

21

bb

aak

Example 11

Compute a) x

b) x

Example 12

Show that

a x

a = 0

Theorem1 : The vector

a x

b is orthogonal to both

a and

b

Theorem2 : If is the angle between

a and

b , then

sinbabxa =


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Two nonzero vectors

a and

b are parallel if and only if

a x

b = 0

The geometry interpretation of theorem 2 can be seen at figure below. If a and

b are represented by directed line segments with the same initial point, then

they determine a parallelogram with base a altitude sinb , and the area

A = bxa)sinb(a =

Example 13

Find a vector perpendicular to the plane that passes through the points

P(1, 4, 6), Q(-2, 5, -1) and R(1, -1, 1)

Example 14

Find the area of the triangle with vertices P(1, 4, 6), Q(-2, 5, -1) nd R(1, -1, 1)

If we apply Theorem 1 and 2 to the standard basis vectors i,j, andkusing

2/= , we obtain

ixj=k jxk= i kx i =j

jx i= - k kxj= - i i xk = - j

Observe that ixj jx i

Theorem 3: If

a ,

b and

c are vectors and k is a scalar, then

i) a x b = - b x a ii) (ka)x b = k( a x b)= a x (kb)iii) a x ( b + c)= a x b + a x c iv) (a + b)x c = a x b + a x c v) a . (b x c) = (a x b) . c

a

b

sinb


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vi) a x ( b x c ) = (a . c) b - ( a . b ) c The product a . (b x c) in (v) is called the scalar triple product of the vectors a, b

and c. We can write the scalar triple product as a determinant

a . (

b x

c)=

321

321

321

ccc

bbbaaa

The geometric significance of the scalar triple product can be seen by

considering the parallelepiped determined by the vectors

a ,

b and

c in the

figure 1. The volume of parallelepiped determined by

V = (Area of Parallelogram) x h = Ah

= cosacxb = )cxb(.a

The volume of the parallelepiped determined by the vectors a,

b andc is the

magnitude of their scalar triple product)cxb(.a

If we discover that the volume of parallelepiped determined by the vectors

a ,

b and

c is zero, then the vectors must lie in the same plane: that is, they are

coplanar

Example 15

Use the scalar triple product to show that the vectors

a = ,

b = , and c= are coplanar

_

h

bx c

Figure 1

c

b

a


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VECTORS

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Exercises

1. Find

PQ and magnitude of

PQ

a) P = (0,0,0) and Q = (3, -4, 10)b) P = (1,-3,2) and Q = (3, -1, 3)

c) P = (0,1,0) and Q = (3, -1, 7)

2. Compute a+

b and a 2

b

a)

a = 2i - 5j + 10k,

b = -i + 2j - 9k

b)

a = i + 2j - 3k,

b = i - j - 3k

c)

a = 2i ,

b = 2j + k

3. Find a unit vector having the same direction as the given vector

a)

a = i + j - k b)

a = -3i + 4j - 13k

c)

a = 7i + 212 j - 212 k

4. Find two unit vectors parallel to the given vectors and write the given vectors

as the product of its magnitude and a unit vector.

a) b)

b) d)

4. Find a vector with the given magnitude and in the same direction as the givenvector.

a) Magnitude 6,

v =

b) Magnitude 10,

v =

c) Magnitude 2,

v =

5 Find a vector

w of magnitude 4that is parallel to vector

+= k3ji2v

(Oct 2008)

6. Determine whether the vectors aand bare parallel

a)

a = ,

b =

b)

a = ,

b =

c)

a = ,

b =

d)

a = i + 2j - k,

b = 3i + 6j - 3k


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e)

a = 2i + j - k,

b = -i - j - 3k

7. Compute

a .

b and the angle between

a and

b

a)

a = ,

b =

b)

a = ,

b =

c)

a = 2i - k ,

b = 4j - k

d)

a = i + 1/3j - 2k ,

b = 2i - 2j + k

e)

a = i + 2j - k ,

b = 3i + 6j - 3k

8. Determine if the vectors are orthogonal.

a)

a = ,

b =

b)

a = ,

b =

c)

a = -4i + 2j - 5k ,

b = i + 6j + 2k

9. Let

+= kjiA 242 ,

++= kjiB and

+= k2

ji2

C

.

a) Determine which two vectors are perpendicular to each other.

b) Determine which two vectors are parallel to each other.

d) Give reason to your answers.

(April 2008)

10. Find what value of b are the vectors and so that they

are coplanar

11. Find the cross product ax

b and verify that it is orthogonal to both aand

b

a)

a = ,

b =

b)

a = ,

b =

c)

a = 2i + j - k,

b =j + 2k

d)

a = i - j + k ,

b = i + j + k

12. If

a = ,

b = find

a x

b and

b x

a

13 If

a = ,

b = , and C = show that

a x(

b xc)

(

a x

b )xc


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14 Find two unit vector orthogonal to both and

15. Find the area of parallelogram with vertices A(1, 2, 3), B(1, 3, 6) and

C(3, 8,6)

16. Find a vector orthogonal to the plane through the point P, Q and R

a) P(1, 0, 0), Q(0, 2, 0) and R(0, 0, 3)

b) P(2, 1, 5), Q(-1, 3, 4) and R(3, 0, 6)

17. Find the volume of parallelepiped determined by vectors

a ,

b ,

c .

a)

a = ,

b = , c =

b)

a = i + j - k ,

b = i - j + k, c = -i + j + k

18.

Use the scalar triple product to verify that the vectors

a = 2i + 3j + k,and

b = i - j , c= 7i + 3j + 2kare coplanar

19. Use the scalar triple product to verify that the points P(1, 0, 1), Q(2, 4, 6) and

R(6, 2, 8) lie in the same plane.

19. The coordinates of the pointsA, B, CandDare (1,2,3), (2,3,5), (4,3,1) and

(2,0,1) respectively. Show that A, B, C andD are coplanar.

(Oct 2008)

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