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Today’s Topics

- Vectors
- We will introduce the concept of vectors, which have many applications throughout physics, and are the most important new mathematical concept used in the course.

Thursday’s Assignment

- Read Ch. 3, except section 8.
- A problem set on HW3 on Ch. 3 will be due next Tuesday.
- The first exam is now scheduled for Thursday, Sept. 20. The calendar in the syllabus posted on CitLearn has been updated.
- You do not need to memorize equations: the essential ones will be provided for the exam.

Quiz: Question 2

- Which of the equations gives the correct relation between the vectors in the figure?
- A.A + B + C = 0
- B. A = B + C
- C. B = A + C
- D. C = A + B
- E. None of these

→ → →

→

→ → →

A

→

B

→ → →

→ → →

→

C

Quiz: Question 1

- Which of the following is a vector?
- A. Mass
- B. Temperature
- C. Distance
- D. Displacement
- E. Speed

Quiz: Question 3

→ → →

- Suppose C = A – B. Under what circumstances is the length of C equal to the sum of the lengths of A and B?
- A. Always
- B. When A and B point in opposite directions.
- C. Never
- D. When A and B are parallel.
- E. When A and B are perpendicular.

→ →

→

→ →

→ →

→ →

Quiz: Question 4

→

- Vector A has a magnitude of 10 and a direction angle θ = 60omeasured counter-clockwise from the +x axis. What are the magnitude and direction angle of the vector – 2A?
- A. – 20, 60o
- B. 20, 240o
- C. 20, – 30o
- D. – 20, 240o
- E. – 20, – 30o

→

→

A

y

θ = 60o

x

Vectors and Scalars

- Scalars are quantities described entirely by a number, with no need to specify a direction – the temperature, for example.
- Vectorsrequire both a magnitude and direction to be fully specified.
- Describing motion in 2 or more dimensions requires vectors.
- Also forces, which must act in some direction, are described by vectors.

Quiz: Question 1

- Which of the following is a vector?
- A. Mass
- B. Temperature
- C. Distance
- D. Displacement
- E. Speed

Displacement Vectors

- The position of a point Brelative to a point Ais given by a displacement vector Dpointing from AtoB.
- This vector tells you how to get from point Ato point B.

→

B

→

D

A

Cartesian Components

Cartesian coordinates are used to label points in a plane.

The lengths of a vector along the two axes are called its Cartesian components.

Dx = 2, Dy = 5.

y

→

Dy

D

x

0

Dx

Polar Coordinates

A vector can also be specified by giving its magnitudeanddirection.

The magnitude is

the length of the

vector: D = |D|.

The direction can

be given by an

angle relative to

an axis. The angle in polar coordinates is measured counterclockwise from the xaxis.

y

→

→

D

θ

x

0

Mathematical Review: Right Triangle

- The sides of a right triangle satisfy the Pythagorean Theorem:
- a2 + b2 = c2

c

b

a

Mathematical Review: Trigonometry

- The ratios of sides of a right triangle define the trigonometric functions.
- sin θ = b/c cosθ = a/c tan θ = b/a
- cscθ = c/b sec θ = c/a cot θ = a/b
- Inverses: θ = asin (b/c) = acos(a/c) = atan(b/a)

c

b

θ

a

Polar Coordinates

→

Find the magnitude and direction of D.

Dx = 2, Dy = 5

D = √ Dx2 + Dy2

= √29 = 5.4

tan θ = 5/2 = 2.5

θ = tan-1 (2.5)

= 68o

y

→

D

θ

x

0

Vector Addition

- Geometrically, two vectors are added by following one to the end, then following the second from that point, and finding the net displacement.
- Components:
- = +

Cx = Ax + Bx

Cy = Ay + By

→

→

→

→

→

→

A

B

C

C

B

A

Quiz: Question 2

- Which of the equations gives the correct relation between the vectors in the figure?
- A.A + B + C = 0
- B. A = B + C
- C. B = A + C
- D. C = A + B
- E. None of these

→ → →

→

→ → →

A

→

B

→ → →

→ → →

→

C

Vectors

- Two vectors, AandB, of length 5and3respectively, lie in a plane, but the directions are unspecified.
- What is the maximum magnitude of A + B?
- |A+B| = 8
- What is the minimum magnitude of A + B?
- |A+ B|=2

→ →

→ →

→ →

→ →

→

→

→

→

→

→

C

B

A

A

B

C

→ →

Scalar Multiple

- Vectors can be multiplied by scalars (numbers).
- Multiplying by a positive number changes the length, not the direction:
- Multiplying by a negative number also changes the direction by 180o:

→

→

→

→

A

2A

A

– A

Quiz: Question 4

→

- Vector A has a magnitude of 10 and a direction angle θ = 60omeasured counter-clockwise from the +x axis. What are the magnitude and direction angle of the vector – 2A?
- A. – 20, 60o
- B. 20, 240o
- C. 20, – 30o
- D. – 20, 240o
- E. – 20, – 30o

→

→

A

y

θ = 60o

x

Quiz: Question 4

→

- Vector A has a magnitude of 10 and a direction angle θ = 60omeasured counter-clockwise from the +x axis. What are the magnitude and direction angle of the vector – 2A?
- A. – 20, 60o
- B. 20, 240o
- C. 20, – 30o
- D. – 20, 240o
- E. – 20, – 30o

→

→

y

A

θ = 240o

10

θ = 60o

x

20

→

– 2A

Vector Difference

→

→

- The vector difference A–B can be formed by adding the vector – B to the vector A.
- A–B can be interpreted as the displacement that takes you from B to A.

→

→

→

→

A – B

→

A

→

B

→

→

→

→

→

– B

Quiz: Question 3

→ → →

- Suppose C = A – B. Under what circumstances is the length of C equal to the sum of the lengths of A and B?
- A. Always
- B. When A and B point in opposite directions.
- C. Never
- D. When A and B are parallel.
- E. When A and B are perpendicular.

→ →

→

→ →

→ →

→

→

→

C

A

B

→ →

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