1. 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

2. Dot product Definition If and, then the dot product of a and b is a  b which is defined by

3. The properties of dot product If a, b, and c are vectors in the same dimensions, and k is scalar, then 1. a  a = 4. ( ka)  b) = k(a  b) = a  (kb) 2. a  b = b  a 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 or

4. 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 . Vector a and b orthogonal (perpendicular) if and only if a  b = 0. 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 .

5. Projection a b v Vector v is called the vector projection of b to a. The magnitude of vector v is called scalar projection of b to a.  For example: Determine the scalar projection and the vector projection of b =  1, 1, 2  to a =  -2, 3, 1 

6. F  Work A constant force F cause a movement of from P to Q. has a deviation vector which is defined by The work of this force is defined as the multiplication of the component of that force along d as the distance of the movement PQ R S 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..

7. Cross product Definition If and, then the cross product of a and b is vector Supported notation : For example If a =  1,3,4  and b =  2,4,-3 , determine a  b.

8. Theorem 5.2 Vector a  b orthogonal either to a or b. ba a  b  Theorem 5.3 If  the angle between vectors a and b (0     ), then  a b For example Determine the area of triangle which vertices are A(1,2,4), B(-2,6,-1), and C(1, 0, 5). The magnitude of cross product a  b equals the area of parallelogram which is determined by vectors a and b.

9. Theorem 5.4 If a, b and c vectors and k scalar, then 1. a  b = -b  a 2.( k a)  b = k (a  b) = a  ( k b) 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 Consequence: Two nonzero vectors a and b paralel if and only if a  b = 0.

10. Scalar triple product: The volume of parallel epipedum which is determined by vectors a, b and c is the value of scalar triple product of b c a b  c

11. 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 .