Chapter 12 – Vectors and the Geometry of Space 12.2 – Vectors 1

Definition - Vector The term vector is used to indicate a quantity that has both magnitude (length) and direction. We denote a vector by a boldface letter (v) or by putting an arrow above the letter ( ) – Vectors2

Notation Vectors always start from an initial point called the tail and continue to the terminal point called the tip. We indicate this by writing v =. Vectors u and v are equal in direction and magnitude so u=v – Vectors3

Definition - Zero Vector The zero vector, denoted by 0, has length zero. It is the ONLY vector with no specific direction – Vectors4

Definition – Vector Addition If u and v are vectors positioned so the initial point of v is at the terminal point of u, then the sum u+v is the vector from the initial point of u to the terminal point of v. Triangle Law Parallelogram Law 12.2 – Vectors5

Visual This visual shows how the Triangle and Parallelogram Laws work for various vectors a and b. This visual shows how the Triangle and Parallelogram Laws work for various vectors a and b – Vectors6

Try This On Your Own Head to toe vector addition – Vectors7

Definition – Scalar Multiplication If c is a scalar and v is a vector, then the scalar multiple cv is the vector whose length is |c| times the length of v and whose direction is the same as v if c>0 and opposite to v if c<0. If c=0 or v=0, then cv= – Vectors8

More on Scalars Real numbers work like scaling factors. Two non-zero vectors are parallel if they are scalar multiples of each other. By the difference u – v of two vectors we mean 12.2 – Vectors9

Constructing u-v Method 1 – Parallelogram Law Draw v and –v and then add it to u. Method 2 – Triangle Law 12.2 – Vectors10

Components We can place the initial point of a vector a at the origin of a rectangular coordinate system. The terminal point of a has the coordinates of the form (a 1,a 2 ) or (a 1,a 2,a 3 ) depending on if our coordinate system is a 2D or 3D one. The components are diagramed and written as follows: 12.2 – Vectors11

Definition – Position Vector The position vector,, is the representation of the vector from the origin to the point P – Vectors12

Representing other vectors Given the points A(x 1,y 1,z 1 ) and B(x 2,y 2,z 2 ), the vector a with representation is 12.2 – Vectors13

Magnitude The magnitude or length is denoted by |v| or ||v|| and obtained by the formulas: 12.2 – Vectors14

Combining Vectors Adding & Subtracting vectors and multiplying a vector by a scalar – Vectors15

Properties of vectors If a, b, and c are vectors in V n and c and d are scalars, then 12.2 – Vectors16

Standard Base Vectors Vectors i, j, and k are called the standard base vectors. They have length 1 and point in the direction of the positive axis – Vectors17

Example 1 – pg. 777 #20 Find a+b, 2a+3b, |a|, |a-b| if a = 2i – 4j + 4k b = 2j - k 12.2 – Vectors18

Definition – Unit Vector A unit vector, u, is a vector whose length is 1. For example, i, j, and k are all unit vectors – Vectors19

Example 2 – pg. 799 # 23 Find a unit vector that has the same direction as the given vector. -3i + 7j 12.2 – Vectors20

Example 3 – pg. 799 #30 If a child pulls a sled through the snow on a level path with a force of 50N exerted at an angle of 38 o above the horizontal, find the horizontal and vertical components of the force – Vectors21

Example 4 – pg. 799 #34 The magnitude of a velocity vector is called speed. Suppose the wind is blowing from the direction N45 o W at a speed of 50 km/h. A pilot is steering a plane in the direction N60 o E at an airspeed (speed in still air) of 250 km/h. The true course, or track, of the plane is the direction of the resultant of the velocity vectors of the plane and wind. The ground speed of the plane is the magnitude of the resultant. Find the true course and the ground speed of the plane – Vectors22

Example 5 – pg. 799 #38 The tension T at each end of the chain has magnitude 25 N. What is the weight of the chain? 12.2 – Vectors23

More Examples The video examples below are from section 12.2 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦ Example 1 Example 1 ◦ Example 3 Example 3 ◦ Example 4 Example – Three Dimensional Coordinate Systems24

Demonstrations Feel free to explore these demonstrations below. Head-to-Toe Vector Addition Vectors in 3D 12.2 – Vectors25