1.
Determine vectors and scalars from these following quantities: weight,
specific heat, density, volume, speed, calories, momentum, energy, distance.
2.
A car moving towards the north as far as 3 miles, then 5 miles to the
northeast. Describe this movement graphically and determine the resultant
displacement vectors graphically and analytically.
3.
Show that the addition of vectors is commutative.
4.
Given a = 3, -2, 1, b = 2, -4, -3, c = -1, 2, 2 determine the length of a,
a+b+c, dan 2a-3b-5c
5.
.
Given a = 2, -1, 1, b = 1, 3, -2, c = -2, 1, -3, and d = 3, 2, 5 determine
scalars k, l, m so that d=ka+lb+mc
Dot product
Definition
If a  a1 , a2 , a3 and b  b1, b2 , b3 , then the dot product of a and b is ab
which is defined by
a  b  a1b1  a2b2  a3b3
The properties of dot product
If a, b, and c are vectors in the same dimensions, and k is scalar,
then
1. a  a = a
2
2. a  b = b  a
4. (ka)  b) = k(a  b) = a  (kb)
5. 0  a = 0
3. a  (b + c) = a  b +a  c
Theorem 5.1
If  is the angle between vectors a and b, then
a b
a  b  a b cos  or
cos  
ab
E.g:
1. If the length of vectors a and b are 3 and 8, respectively, and
the angle between those two vectors is /3, determine ab.
2. Determine the angle between vectors a = 2,2,-1 and b = 5,3,2.
Vector a and b orthogonal (perpendicular) if and only if a  b = 0.
E.g:
1. Show that 2i – 2j + k is perpendicular to 5i + 4j – 2k.
2. Determine the value of x so that vector a = 1,2,1 and b = 1,0,
x  formed an angle which magnitude is 60.
Projection
Vector v is called the vector
projection of b to a.
b
The magnitude of vector v is called

scalar projection of b to a.
v
a
proyeksi skalar : v 
a b
a
 a b  a a b
a b
proyeksi vektor  

a

a
2
 a  a
aa
a


For example:
Determine the scalar projection and the vector projection of
b = 1, 1, 2 to a = -2, 3, 1
Work
R
A constant force F cause a movement
of from P to Q. has a deviation vector
F
which is defined by d  PQ
The work of this force is defined as the

P
S
Q
multiplication of the component of that
force along d as the distance of the
movement
W   F cos   d  F  d
For example:
A force F = 3i + 4j +5k cause the movement of a particle from P(2,1,0)
to Q(4,6,2). Determine the work which is done by F..
Cross product
Definition
If a  a1 , a2 , a3 and b  b1, b2 , b3 , then the cross product of a and b is
vector
a  b  a2b3  a3b2 , a3b1  a1b3 , a1b2  a2b1
Supported notation :
i
j
a  b  a1 a2
b1 b2
k
a
a3  2
b2
b3
a3
b3
i
a1 a3
b1
b3
For example
If a = 1,3,4 and b = 2,4,-3, determine a  b.
j
a1 a2
b1
b2
k
ab
Theorem 5.2
Vector a  b orthogonal either to a or b.

a
b
Theorem 5.3
If  the angle between vectors a and b (0   ),
b
b sin 

then
a  b  a b sin 
a
The magnitude of cross product a  b equals the area of
parallelogram which is determined by vectors a and b.
For example
Determine the area of triangle which vertices are A(1,2,4), B(-2,6,-1),
and C(1, 0, 5).
Consequence:
Two nonzero vectors a and b paralel if and only if a  b = 0.
Theorem 5.4
If a, b and c vectors and k scalar, then
1. a  b = -b  a
2. (ka)  b = k(a  b) = a  (kb)
3. a  (b + c) = a  b + a  c
4. (a + b)  c = a  c + b  c
5. a  (b  c) = (a  b)c
6. a ( b  c) = (ac)b – (ab)c
a1 a2
Scalar triple product:
a  (b  c)  b1
c1
b2
c2
a3
b3
c3
The volume of parallel epipedum which is determined by vectors a,
b and c is the value of scalar triple product of
V  a  (b  c)
bc
a
c
b
E.g:
Determine the volume of a parallel epipedum which the sides are a, b,
and c which are defined as a = i + 2k, b = 4i + 6j + 2k, and c = 3i +3j – 6k
Show that these following vectors are in the same plane: a = 1,4,-7, b =
2,-1,4 and c = 0,-9,18.