AP Physics Chapter 3 Vector

AP Physics Turn in Chapter 2 Homework, Worksheet, & Lab Take quiz Lecture Q&A

Vector and Scalar Vector: Scalar: Magnitude: How large, how fast, … Direction: In what direction (moving or pointing) Representation depends on frame of reference Scalar: Magnitude only No direction Representation does not depend on frame of reference

Examples of vector and scalar Vectors: Position, displacement, velocity, acceleration, force, momentum, … Scalars: Mass, temperature, distance, speed, energy, charge, …

Vector symbol Vector: bold or an arrow on top Scalar: regular Typed: v and V or Handwritten: Scalar: regular v or V v stands for the magnitude of vector v.

Adding Vectors Graphical Analytical (by components) Head-to-Tail (Triangular) Parallelogram Analytical (by components)

Graphical representation of vector: Arrow An arrow is used to graphically represent a vector. The length of the arrow represents the magnitude of the vector. head b a tail The direction of the arrow represents the direction of the vector. When comparing the magnitudes of vectors, we ignore directions. Vector a is smaller than vector b because a is shorter than b.

Equivalent Vectors Two vectors are identical and equivalent if they both have the same magnitude and are in the same direction. They do not have to start from the same point. (Their tails don’t have to be at the same point.) A B C A, B and C are all equivalent vectors.

Negative of Vector Vector -A has the same magnitude as vector A but points in the opposite direction. If vector A and B have the same magnitude but point in opposite directions, then A = -B, and B = -A A -A -A

Adding Vectors: Head-to-Tail Head-to-Tail method: B A Example: A + B Draw vector A Draw vector B starting from the head of A A+B B The vector drawn from the tail of A to the head of B is the sum of A + B. A Make sure arrows are parallel and of same length.

A+B=B+A B A A + B A B+A B How about B + A? What can we conclude? A+B B

A+B+C B A C C B A+B+C A Resultant vector: from tail of first to head of last.

A-B=A+(-B) A B A A-B -B Draw vector A Draw vector -B from head of A. The vector drawn from the tail of A to head of –B is then A – B.

What Are the Relationships? c b a a b c

Magnitude of sum A + B = C |A – B|  C  A + B max. c min. c A and B in same direction A and B in opposite direction A and B at some angle max. c min. c |A – B|  C  A + B

Adding Vectors: Parallelogram A+B Draw the two vectors from the same point Construct a parallelogram with these two vectors as two adjacent sides B A+B The sum is the diagonal vector starting from the same tail point. A B Advantage: No need to measure length. A

N 35o Example W E S Vector a has a magnitude of 5.0 units and is directed east. Vector b is directed 35o west of north and has a magnitude of 4.0 units. Construct vector diagrams for calculating a + b and b – a. Estimate the magnitudes and directions of a + b and b – a from your diagram.

Solution N -a b-a b a+b W E a S Using ruler and protractor, we find: a+b: 4.3 unit, 50o North of East b-a: 8.0 unit, 66o West of North

Vector Components y a ay x ax Drop perpendicular lines from the head of vector a to the coordinate axes, the components of vector a can be found: y a ay  x ax  is the angle between the vector and the +x axis. ax and ay are scalars.

Finding components of a vector Resolving the vector Decomposing the vector

Vector magnitude and direction y ay a Vector magnitude and direction  ax x The magnitude and direction of a vector can be found if the components (ax and ay) are given: (for 3-D)  is the angle from the +x axis to the vector.

Example A ship sets out to sail to a point 120 km due north. Before the voyage, an unexpected storm blows the ship to a point 100 km due east of its starting point. How far, and in what direction, must it now sail to reach its original destination?

A ship sets out to sail to a point 120 km due north A ship sets out to sail to a point 120 km due north. Before the voyage, an unexpected storm blows the ship to a point 100 km due east of its starting point. How far, and in what direction, must it now sail to reach its original destination? Solution N  c a =120 It must sail 156 km at 39.8o West of North to reach its original destination.  E b = 100

Practice: What are the magnitude and direction of vector y 5  x 4

Practice: What are the components of a vector that has a magnitude of 12 units and makes an angle of 126o with the positive x direction? y 126o x

Unit vectors Unit vector: magnitude of exactly 1 and points in a particular direction. i or x direction: y direction: j or z direction: k or

Vector components and expression Any vector can be written in its components and the unit vectors:

Terminology axi is the vector component of a. ax is the (scalar) component of a.

Example Express the following vector in component and unit vector form. y a=12.0 units ay =30o ax x

Adding Vectors by Components x y ry r by b Adding Vectors by Components ay bx a rx ax When adding vectors by components, we add components in a direction separately from other components. 2-D 3-D Component form:

Example The minute hand of a wall clock measures 10 cm from axis to tip. What is the displacement vector of its tip (a) from a quarter after the hour to half past, (b) in the next half hour, and (c) in the next hour? • • •

Solution or magnitude and direction form. y rBC Try b) and c) x rAB B (0,-10)

Two vectors are given by a = 4i – 3j + k and b = -i + j + 4k. Find: a + b a – b a vector c such that a – b + c = 0 Practice

Practice: 54-19 The two vectors a and b in Fig Practice: 54-19 The two vectors a and b in Fig. 3-29 have equal magnitudes of 10.0 m and the angels are 1 = 30o and 2 = 105o. Find the (a) x and (b) y components of their vector sum r, (c) the magnitude of r, and (d) the angle r makes with the positive direction of the x axis. y b b=105o+30o=135o r 2 a 1 a x

54-19 (Continued)

Vector Multiplication More: Scalar (aka dot or inner) product: a  b Vector (aka cross) product: a  b We cannot write ab if a and b are vectors. But we still can write 2a since 2 is a scalar.

Scalar product: ab  (phi) is the angle between vector a and b. : angle between vector and +x axis : angle between two vectors Scalar product: ab b a   (phi) is the angle between vector a and b.  is always between 0o and 180o. (0o    180o) The scalar product is a scalar  It has no direction. What if the two vectors are perpendicular to each other?

Physical meaning of ab ba is the projection of b onto a. a b  ba Also ab is the project of a onto b. ab a b 

Properties of scalar product ab = ba ii = jj = kk = 1 ij = ji = jk = kj = ki = ik = 0 aa = a2 ab = 0 if a  b 6. ab = axbx+ayby+azbz

Angle between two vectors When we know magnitudes and : When we know the components: Put together:

Example: a. Determine the components and magnitude of r = a – b + c if a = 5.0i + 4.0j – 6.0k, b = -2.0i + 2.0j + 3.0k, and c = 4.0i + 3.0 j + 2.0k. b. Calculate the angle between r and the positive z axis. z C • x O 5 y 11 A -7 B

Another approach We are looking for the angle between r and any vector in the z direction. Let’s choose the unit vector in the z direction, k

Practice: 55-43 For the vectors in Fig Practice: 55-43 For the vectors in Fig. 3-35, with a = 4, b =3, and c =5, calculate B b a c 5 3 4 A  

Approach 2 y b c b -4 x a -3 c

Approach 3 b a c 5 3 4

Approach 4 d = -c b a c 5 3 4 ce e = -b da

Vector Product: c = a  b Magnitude of c is: c is a vector, and it has a direction given by the right-hand-rule (RHR): Place the vectors a and b so that their tails are at the same point. Extend your right arm and fingers in the direction of a. Rotate your hands along your arm so that you can flap your fingers toward b through the smaller angle between a and b. Then Your outstretched thumb points in the direction of c.

Properties of cross product b  a = - (a  b) a  b is  a, and a  b is  b a  a = 0

Practice: Three vectors are given by a = 3. 0i + 3. 0j – 2. 0k, b = -1 Practice: Three vectors are given by a = 3.0i + 3.0j – 2.0k, b = -1.0i –4.0j + 2.0k, and c = 2.0i + 2.0j + 1.0k. Find (a) a • (b  c), (b) a • (b + c), and (c) a  (b + c).

Pg52-53P (2)

Pg52-53P (3)

Law of cosine c b C a

Law of Sine or C b a B A c