Chapter 12 Vectors A. Vectors and scalars B. Geometric operations with vectors C. Vectors in the plane D. The magnitude of a vector E. Operations with plane vectors F. The vector between two points G. Vectors in space H. Operations with vectors in space I. Parallelism J. The scalar product of two vectors

Opening Problem An airplane in calm conditions is flying at 800 km/hr due east. A cold wind suddenly blows from the south-west at 35 km/hr, pushing the airplane slightly off course. Things to think about: a. How can we illustrate the plane’s movement and the wind using a scale diagram? b. What operation do we need to perform to find the effect of the wind on the airplane? c. Can you use a scale diagram to determine the resulting speed and direction of the airplane?

Vectors and Scalars Quantities which have only magnitude are called scalars. Quantities which have both magnitude and direction are called vectors.

The speed of the plane is a scalar. It describes its size or strength. The velocity of the plane is a vector. It includes both its speed and also its direction. Other examples of vector quantities are: accelerationforce Displacementmomentum

Directed Line Segment Representation We can represent a vector quantity using a directed line segment or arrow. The length of the arrow represents the size or magnitude of the quantity, and the arrowhead shows its direction.

For example, farmer Giles needs to remove a fence post. He starts by pushing on the post sideways to loosen the ground. Giles has a choice of how hard to push the post and in which direction. The force he applies is therefore a vector. If farmer Giles pushes the post with a force of 50 Newtons (N) to the north-east, we can draw a scale diagram of the force relative to the north line. 45 o N 50N Scale: 1 cm represents 25 N

Vector Notation

Geometric vector equality Two vectors are equal if they have the same magnitude and direction. Equal vectors are parallel and in the same direction, and are equal in length. The arrows that represent them are translations of one another.

Geometric negative vectors AB and BA have the same length, but they have opposite directions. A A B B

Geometric operations with vectors A typical problem could be: A runner runs east for 4 km and then south for 2 km. How far is she from her starting point and in what direction? 4 km 2 km x km N S WE

Geometric vector addition Suppose we have three towns P, Q, and R. A trip from P to Q followed by a trip from Q to R has the same origin and destination as a trip from P to R. This can be expressed in vector form as the sum PQ + QR = PR. P R Q

“head-to-tail” method of vector addition To construct a + b: Step 1: Draw a. Step 2: At the arrowhead end of a, draw b. Step 3: Join the beginning of a to the arrowhead end of b. This is vector a + b.

Given a and b as shown, construct a + b. a b

a b a b a + b

THE ZERO VECTOR The zero vector 0 is a vector of length 0. For any vector a: a + 0 = 0 + a = a a +(-a) =(-a) + a = 0.

Find a single vector which is equal to: a. BC + CA b. BA + AE + EC c. AB + BC + CA d. AB + BC + CD + DE A ED C B

A ED C B BC + CA = BA

A ED C B BA + AE + EC = BC AB + BC + CA = AA = 0 AB + BC + CD + DE = AE

Geometric vector subtraction To subtract one vector from another, we simply add its negative. a – b = a +(-b) a b a b -b a - b

For r, s, and t shown, find geometrically: a. r – s b. s – t – r r s t

r – s r s -s r – s

s – t – r r s t -t s – t – r

For points A, B, C, and D, simplify the following vector expressions: a. AB – CB b. AC – BC – DB

a. Since –CB = BC, then AB – CB = AB + BC = AC. b. Same argument as part a. AC – BC – DB = AC + CB + BD = AD

Vector Equation Whenever we have vectors which form a closed polygon, we can write a vector equation which relates the variables.

Find, in terms of r, s, and t: a. RS b. SR c. ST O T S R r t s

a. Start at R, go to O by –r then go to S by s. Therefore RS = -r + s = s – r. b. Start at S, go to O by –s then go to R by r. Therefore SR = -s + r = r – s. c. Start at S, go to O by –s then go to T by t. Therefore ST = -s + t = t – s.

Geometric scalar multiplication If a is a vector and k is a scalar, then ka is also a vector and we are performing scalar multiplication. If k > 0, ka and a have the same direction. If k < 0, ka and a have opposite directions. If k = 0, ka = 0, the zero vector.

Given vectors r and s, construct geometrically: a. 2r + s b. r – 3s r s

a. 2r + s r s 2r s 2r + s

b. r – 3s r s r -3s r – 3s

Vectors in the plane In transformation geometry, translating a point a units in the x-direction and b units in the y-direction can be achieved using the translation vector

Base unit vector All vectors in the plane can be described in terms of the base unit vectors i and j.

a. 7i + 3j 7i 3j

b. -6i c. 2i – 5j d. 6j e. -6i + 3j f. -5i – 5j

The magnitude of a vector v1v1 v2v2 v

Unit vectors A unit vector is any vector which has a length of one unit. are the base unit vectors in the positive x and y-directions respectively.

Find k given that is a unit vector.

Knowing that is a unit vector, then

Operations with plane vectors a a1a1 a2a2 a 1 +b 1 b b1b1 b2b2 a 2 +b 2 a+b

Check your answer graphically.

Algebraic negative vectors

Algebraic vector subtraction

Algebraic scalar multiplication

If p = 3i – 5j and q =-i – 2j, find |p – 2q|.

p – 2q = 3i – 5j – 2(-i – 2j) = 3i – 5j + 2i + 4j = 5i – j

The vector between two points O A B a1a1 b1b1 a2a2 b2b2

Given points A(-1, 2), B(3, 4), C(4, -5) and O(0, 0), find the position vector of: a. B from O b. B from A c. A from C

[AB] is the diameter of a circle with center C(-1, 2). If B is (3, 1), find: a. BC b. the coordinates of A.

Vectors in space

3-D point plotter: demos #72 content disk

Recall the length of the diagonal of a rectangular prism (box), or if it’s easier, call it “3-D Pythagorean Theorem”

Illustrate the points: a. A(0, 2, 0) b. B(3, 0, 2) c. C(-1, 2, 3)

a. A(0, 2, 0)

b. B(3, 0, 2)

c. C(-1, 2, 3)

The vector between two points

If P is (-3, 1, 2), Q is (1, -1, 3), and O is (0, 0, 0), find: a. OP b. PQ c. |PQ|

Vector Equality

ABCD is a parallelogram. A is (-1, 2, 1), B is (2, 0,-1), and D is (3, 1, 4). Find the coordinates of C.

A(-1,2,1) B(2,0,-1) D(3,1,4) C(a,b,c)

Operations with vectors in space

Properties of vectors

Two useful rules are:

Using the one of the properties of vectors, we know that |ka| = |k| |a| Therefore |2a| = 2|a|

Find the coordinates of C and D:

Parallelism

Parallelism Properties

Unit vectors

If a = 3i - j find: a. a unit vector in the direction of a b. a vector of length 4 units in the direction of a c. vectors of length 4 units which are parallel to a.

a. a unit vector in the direction of a

b. a vector of length 4 units in the direction of a

c. vectors of length 4 units which are parallel to a.

Find a vector b of length 7 in the opposite direction to the vector

The scalar product of two vectors There are two different types of product involving two vectors: 1. The scalar product of 2 vectors, which results in a scalar answer and has the notation v●w (read “v dot w”). 2. The vector product of 2 vectors, which results in a vector answer and has the notation vΧw (read “v cross w”).

Scalar product

ALGEBRAIC PROPERTIES OF THE SCALAR PRODUCT