6 (Vectors)

(Force)

(Vectors in the Plane)

6.1

AB A B AB

AB

6.1

6.1 P(x1, y1) Q(x2, y2) PQ v

v = < x2 x1, y2 - y1> (Magnitude) PQ | PQ |

| PQ | = | v | = 212

212

)yy()xx(

6.1 A = (1, 2) B = (3, 4) AB | AB |

AB = < 3 1, 4 2 > = < 2, 2 >

| AB | = 22 22 = 2 2

6.2 u v u = v u v

A

B

AB

6.1

80

6.3 (Zero vector) 0 = < 0, 0 >

6.4 (Unit Vector) 1

< 1, 0 >, < 0, 1 >, < 2

1 ,2

1 >

6.1 u = < a, b > v = < c, d > u = v a = c b = d

6.5 u = < u1, u2 > v = < v1, v2 >

1. u + v = < u1 + v1, u2 + v2 >

2. k u = < ku1, ku2 > k

3. u v = u + ( v ) = < u1 v1, u2 v2>

6.2 u , v w a b

1. u + v = v + u 2. ( u . v ) + w = u + ( v + w ) 3. u + 0 = u 4. u + ( u ) = 0 5. 0 u = 0 6. 1 u = u 7. a ( u + v ) = a u + a v 8. |a u | = |a|| u |

6.2 u = < 2, 3 > v = < 1, 2 > 1. u + v 2. u v

3. 3 u 4. | 2 v |

1. u + v = < 2 + 1, 3 + 2 > = < 1, 5 >

2. u v = < 2 1, 3 2 > = < 3, 1 >

3. 3 u = < 3(2), 3(3) > = < 6, 9 >

81

4. |2 v | = |2|| v | = 2 22 21 = 2 5 6.2 (Standard Unit Vector)

6.6 i = < 1, 0 > j = < 0, 1 >

6.3 v v i j v = < a, b >

v = < a, 0 > + < 0, b > = a < 1, 0 > + b < 0, 1 > = a i + b j

v = < 3, 4 > v = 3 i 4 j

6.4 v v 0

v = |v|

v

6.3 v = < 3, 4 > v

| v | = 22 43 = 5

v |v|

v = 5

i3 + 5

j4

6.3

6.7 p p

6.4 y = 3

x 5 + 3

2

(1, 1)

y = 3

x 5 + 3

2

dx

dy = 3

x5 4

(1, 1) = 3

5

(1, 1) v = 3 i + 5 j | v | = 259 = 34 (1, 1)

82

u = |v|

v = 34

i3 + 34

j5

(1, 1) u = 34

i3 34

j5

(1, 1) = 5

3

(1, 1) p = 5 i 3 j | p | = 925 = 34 (1, 1) q =

|p|

p = 34

i5 34

j3

(1, 1) - q = |p|

p = 34

i5 + 34

j3

6.4 (Dot Product)

6.8 u = < u1, u2 > v = < v1, v2 > u v

u . v u . v = u1v1 + u2v2

6.5 u v u v = | u || v |cos u v 0

u v

= cos-1

|v||u|

vu

6.5 u = < 1, 2 > v = < 2, 1 > 1. u v 2. u v

1. u v = 1(2) + 2(1) = 4

2. u v =

= cos-1

|v||u|

vu

= cos-1

55

4 = cos-1

5

4

6.6 u , v w k 1. u v = v u 2. (k u ) v = u (k v ) = k( u v )

83

3. u ( v + w ) = u v + u w 4. u u = | u |2 5. 0 u = 0

6.7 u v u . v = 0

6.6 u v

1. u = < 3, 1 >, v = < 2, 6 > 2. u = < 2, 1 >, v = < 1, 1 >

1. u v = 3(2) + (1)6 = 0 u v

2. u v = 2(1) + 1(1) = 1 u v

6.5 (Vector in Space)

2 3

6.1 OP 6.2 PQ

6.1 OP = x i + y j + z k i = < 1, 0, 0 > , j = < 0, 1, 0 > , k = < 0, 0, 1 >

6.2 PQ = (x2 x1) i + (y2 y1) j + (z2 z1) k

X

Z

Y

Q(x2, y2, z2)

P(x1, y1, z1) X

Z

Y

P(x, y, z) O

84

6.9 u = u1 i + u2 j + u3 k v = v1 i + v2 j + v3 k c

1. u + v = (u1+ v1) i + (u2+ v2) j + (u3 + v3) k

2. u v = (u1 v1) i + (u2 v2) j + (u3 v3) k

3. c u = (cu1) i + (cu2) j + (cu3) k 4. | u | = 232221 uuu

5. u v = u1v1 + u2v2 + u3v3

6.7 u = 2 i + j - k v = i + j + 2 k 1. u + v

2. u - v 3. | u | | v |

4. u v 5. u v 6. u v

1. u + v = (2 + 1) i + (1 + 1) j + (1 + 2) k = 3 i + 2 j + k

2. u v = (2 1) i + (1 1) j + (1 2) k = i 3 k

3. | u | = 222 )1(12 = 6 | v | = 222 211 = 6

4. u |u|

u = 6

2i +

6

1j

6

1 k

v |v|

v = 6

1i +

6

1j +

6

2 k

5. u v = (21) + (11) + ( 12) = 2 + 1 2 = 1

6. u v = = cos-1

|v||u|

vu

= cos-1

66

1

= cos-1

6

1

85

6.6

6.10 u = u1 i + u2 j + u3 k v = v1 i + v2 j + v3 k u v u v

u v = (u2v3 u3v2) i (u1v3 u3v1) u2 j + (u1v2 u2v1) k

6.8 u = u1 i + u2 j + u3 k v = v1 i + v2 j + v3 k

u v = 321

321

vvv

uuu

kji

6.9 u = u1 i + u2 j + u3 k v = v1 i + v2 j + v3 k u v u v

6.8 u = 2 i j + 3 k v = i + j + k u v

u v = 111

312

kji

= (1 3) i (2 3) j + (2 + 1) k = 4 i + j + 3 k u v u v = 4 i + j + 3 k

86

1. AB | AB | A B 1.1 A = (3, 2) B = (4, 1) 1.2 A = (0, 2) B = (3, -4)

2. v 2.1 v = < -3, 4 > 2.2 v = < 2, 5 > 2.3 v = i + j 2.4 v = 3 i + 4 j 2.5 v = 2 i + j - 3 k 2.6 v = 3 i + 2 j + k

3.

3.1 y = x2 + 2x + 1 (1, 4)

3.2 y = x3+ 2 (-1, 1)

3.3 y = x4 + 3x2 + 2 (1, 6)

3.4 y = 2x2 - 3x (2, 2)

4. u v 4.1 u = < 3, 1 >, v = < 2, 6 > 4.2 u = < 2, 1 >, v = < 1, 1 > 4.3 u = 2 i + 3 j - 4 k , v = 3 i + j + k 4.4 u = 2 i + j + k , v = 3 i - 2 j - 4 k

5. u = 3 i + 2 j + 2 k v = 2 i - 2 j + k

5.1 u + v 5.2 u - v 5.3 u v 5.4 u v 5.5 u v

87

6. u v 6.1 u = i + j + k v = 2 i + j + 2 k 6.2 u = 2 i + 3 j - k v = i + 2 j + 3 k 6.3 u = 3 i + 2 j + k v = i + 2 j + 3 k 6.4 u = i + 2 k v = 2 j + k