1. Geometric and Algebraic

2.  A vector is a quantity that has both magnitude and direction.  We use an arrow to represent a vector.  The length of the arrow is the magnitude and the arrowhead indicates the direction of the vector.

3.  The points on a line joining the two points form what is called a line segment.  If we order the points so that they proceed from one point to another, we have a directed line segment or a geometric vector, denoted by the two points with an arrow on top of it.  The beginning point is called the initial point.  The ending point is called the terminal point.

4.  The vector v whose magnitude is 0 is called the zero vector, 0. The zero vector is assigned no direction.  Two vectors are equal if they have the same magnitude and direction. (Both must be the same.)  See example on p. 619.

5.  To find the sum of two vectors we position the vectors so that the terminal point of the first vector coincides with the initial point of the second vector.  Vector addition is commutative  v + w = w + v Vector addition is also associative u + (v + w) = (u + v) + w

6.  The zero vector has the property that  v + 0 = 0 + v = v  If v is a vector, then –v is the vector having the same magnitude as v, but whose direction is opposite to v.  v + (-v) = 0

7.  If  is a scalar and v is a vector, the scalar product  v is defined as follows: 1. If  > 0, the product  v is the vector whose magnitude is  times the magnitude of v and whose direction is the same as v. 2. If  <0, the product  v is the vector whose magnitude is |  times the magnitude of v and whose direction is the opposite that of v. 3. If a = 0 or if v = 0, then  v = 0.

8.  0v = 0 1v = v-1v = -v  (  v =  v +  v (  v =  v +  v   v)=(  v

9.  p. 628 #s 2 – 8 even

10.  If v is a vector, we use the symbol ||v|| to represent the magnitude. ||v|| is the length of a directed line segment.

11.  If v is a vector and if  is a scalar, then  (a) ||v|| ≥ 0(b) ||v|| = 0 iff v = 0  (c) ||-v|| = ||v||(d) ||  v|| = |  | ||v||

12.  A vector u for which the magnitude of vector u is equal to one is called a unit vector.

13.  An algebraic vector is represented by an ordered pair written in brackets. where a and b are real numbers (scalars) called the components of the vector v.

15.  Find the position vector of the vector

16.  Two vectors v and w are equal iff their corresponding components are equal. We call a and b the horizontal and vertical components of v, respectively.

20.  For any nonzero vector v, the vector  is a unit vector that has the same direction as v.

21.  Find a unit vector in the same direction as  v = 2i – j  We find ||v|| first.

22.  If a vector represents the speed and direction of an object, it is called a velocity vector.  If a vector represents the direction and amount of a force acting on an object, it is called a force vector.  The following helps to find the velocity and force vectors.

23.  A vector written in terms of magnitude and direction is  v = ||v|| (cos  i + sin  j)  where  is the angle between v and i.

24.  A child pulls a wagon with a force of 40 pounds. The handle of the wagon makes an angle of 30 o with the ground. Express the force vector F in terms of i and j.  The magnitude of the Force is given as 40 pounds.  Now express the vector in terms of magnitude and direction

26.  If two forces simultaneously act on an object the vector sum is known as the resultant force. An application of this concept is static equilibrium. An object is said to be in static equilibrium if (1) the object is at rest and (2) the sum of all forces acting on the object is zero.

27.  A weight of 1000 pounds is suspended from two cables as shown in the figure on p. 629 problem 59. What is the tension of the two cables.

28.  On-line Example On-line Example