Introduction to Vectors

UNIT 1

What is a Vector?

• A vector is a directed line segment, can be represented as AB or AB, where A and B are the two endpoints of the line segment.

• Directed means that the vector has a direction.

• QUESTION: Which direction is implied for vector AB?

Vectors: Example

A

B

Vector Quantity

• There are exactly two properties that completely characterize a vector:1) Direction – which way does the vector point?2) Magnitude – the length of the vector, written as |XY|

for vector XYTogether the direction and magnitude define the vector quantity.

• QUESTION: What are some examples of vectors that we are already familiar with?

Equal Vectors

• Two vectors are equal vectors if they have both the same direction and magnitude

Three cars on the road are driving in the same direction at the same speed:

Do they have equal velocity vectors?

Opposite Vectors: An opposite vector is a vector with the same magnitude as the original, but opposite direction:

A

B

In this illustration, AB and BA are opposite vectors.

Vector Sum

• A vector sum A + B is defined as a vector that results from placing the initial point of vector B at the terminal point of vector A: the vector with the same initial point as A and the same terminal point as B is the vector sum.

A

B

A + B

Parallelogram Rule

• Another method, called the parallelogram rule, to find A + B is to place vectors A and B so that their initial points coincide, then complete a parallelogram that has A and B as two adjacent sides. The diagonal of the parallelogram with the same initial points is the vector sum:

Parallelogram Rule Example

A

B

A + B

The two methods side by side:

A

B

A + BA

B

A + B

The two methods give identical results for the vector sum

QUESTION

• Two forces of 15 newtons and 22 newtons act at a point in the plane. If the angle between the forces is 100°, find the magnitude of the resultant force:

22

15

100°

QUESTION

• Two forces of 15 newtons and 22 newtons act at a point in the plane. If the angle between the forces is 100°, find the magnitude of the resultant force:

22

15=80

Z 22

100°

Scalar Product

• A real number k and a vector U create the vector kU which has the magnitude |k| times the magnitude of U . kU has the same direction as U if k>0, and the opposite direction if k < 0:– So vector 2U would be twice the length of U and

would point in the same direction as U does:

Scalar Product Examples

U

2U

3U

-U

-2U

-3U

Position Vectors

• A position vector is a vector with its initial point at the origin and with its endpoint at (a, b). It is written <a, b>, so U = <a, b> below:

(a, b)

U

X axis

Y axis

(0, 0)

Direction Angle of <a, b>

• The direction angle is the positive angle between the x-axis and a position vector:

(a, b)

U

X axis

Y axis

(0, 0)

Direction Angle

Direction Angle of <a, b> (cont)

(a, b)

U

X axis

Y axis

(0, 0)

The direction angle θ satisfies tan θ = b/a, where a ≠ 0:

b

a

Magnitude of vector <a, b>

• The magnitude of vector <a, b> is given by |U| =

(a, b)

U

X axis

Y axis

(0, 0)

b

a

QUESTION

• What is the direction angle and magnitude of vector U=<3, -2>?

(3, -2)U

X axis

Y axis

(0, 0)

QUESTION

• What is the direction angle and magnitude of vector <3, -2>?• ANSWER: Direction angle =

326.3°, magnitude =

Horizontal and Vertical Components of a Vector

• The horizontal and vertical components of a vector U are given by:a = |U|cos θ b = |U|sin θ

(a, b)

U

X axis

Y axis

(0, 0)

b = |U|sin θ

a = |U|cos θ

QUESTION

• Calculate the vertical and horizontal components of a vector with direction angle of 40° and a magnitude of 25.

(a, b)

|U|=25

X axis

Y axis

(0, 0)

= 40

QUESTION

• Calculate the horizontal and vertical components of a vector with direction angle of 40° and a magnitude of 25.• ANSWER: x = 19.2, y = 16.1 or

<19.2, 16.1>

Vector Operations Overview

• <a, b> + <c, d> = <a + c, b + d>• k*<a, b> = <ka, kb>• If A = <a1, a2>, then –A = <-a1, -a2>• <a, b> - <c, d> = <a, b> + -<c, d> OR <a, b> + <-c, -d> = <a – c, b – d>

Vector Operations

• For any real numbers a, b, c, and d:<a, b> + <c, d> = <a + c, b + d>

(a, b)

X axis

Y axis

(0, 0)(c, d)

(a+c, b+d)

Vector Operations

• For any real numbers a, b, c, and d:<a, b> + <c, d> = <a + c, b + d><3, 4> + <4, -2> = <7, 2>

(3, 4)

X axis

Y axis

(0, 0)(4, -2)

(7, 2)

Vector Operations (cont)

B

If U = <a1, a2>

Then –U = <-a1, -a2>

Vector Operations

• Scalar multiplication: k*<a,b> = <ka, kb>• Examples:

-3*<4, 7> = <-12, -21>6*<1, 2> = <6, 12>0*<3, 5> = <0, 0>

Vector Operations

(0, 0)

(3, 4)

(0, 0)

(6, 8)U

2U

QUESTION

• Consider the vectors shown in the following figure, and perform the operations:

UV

U + V(4, 3)

X axis

Y axis

(0, 0)

(-2, 1)

(x, y) a)U + Vb)-2Uc) 4U – 3V

Vector Subtraction

A

B

-A

-B

• Vector subtraction is the inverse operation of vector addition and is defined as adding the negative vector:So we have B – A = B + (-A) for all vectors A, BTherefore (see below) B – A = C

C

Vector Subtraction

• <a, b> - <c, d> = <a, b> + -<c, d> OR <a, b> + <-c, -d> = <a – c, b – d>• Examples:

<3, 6> - <4, 4> = <-1, 2><6, -4> - <-2, -5> = <8, 1><0, 1> - <4, 10> = <-4, -9>

Vector Subtraction QUESTIONS

• QUESTION: Express A as a difference of two vectors.

• QUESTION: Express B as a difference of two vectors.

A

B

C

Vector Notation Conventions

• Unit Vectors: i = <1, 0>, j=<0, 1>• i, j Form for Vectors: If v = <a, b>, then v = ai + bj

Unit Vectors

(5, 3)

U

X axis

Y axis

(0, 0) i ii ii

j

j

j

i j

U = 5i + 3j

QUESTION

• Write the vector in the form ai + bj:

<-5,8>

QUESTION

• Write the vector in the form ai + bj:

<-5,8>Answer: If v=<a,b>, then:V=ai + bj, so V = -5i + 8j

Dot Product

• The dot product of the two vectors U = <a, b> and V = <c, d> is denoted by U•V, read “U dot V,” is given by U•V = ac + bd.

• Examples: <2, 3>•<4, -1> = <6, 4>•<-2, 3> =

Geometric Interpretation of the Dot Product

• If θ is the angle between two nonzero vectors U and V, where 0° < θ < 180°, then

U•V = |U|*|V|cos θ• Example:

<2, 0>•<3, 3> = 6 using <a,b>•<c,d>=ac+bdBut it is also true using U•V = |U|*|V|cos θ

Geometric Interpretation of the Dot Product

(3, 3)

U

X axis

Y axis

(0, 0)

(2, 0)

V

U • V = |U||V|cos θ = (3

θ=45

Properties of the Dot Product

U•V = V•UU•(V+W)=U•V + U•W(U + V)•W = U•W + V•W(kU)•V = k(U•V) = U•(kV)0•U = 0U•U = |U|2

The Dot Product Can Be Positive, Zero, or Negative

θθ < 90: Positive dot product

θ

θ

θ > 90: Negative dot product

θ = 90: Zero dot product