Unit Vectors Walker Ch' 3 Sect 34

1
Unit Vectors (Walker Ch. 3 Sect 3-4)

• A unit vector is a dimensionless vector with a
  magnitude of exactly 1.
• Unit vectors are used to specify a direction and
  have no other physical significance
• Irony A unit vector has no units!!!! ?

2
Unit Vectors, cont.

• The symbols
• or
• represent unit vectors in the x, y and z
  directions, respectively.
• They form a set of mutually perpendicular
  vectors.

3
Unit Vectors in Vector Notation

• is the same as Ax and is the
  same as Ay etc.
• The complete vector can be expressed as
• or
• The units are contained in the magnitude (i.e.,
  Ax) and the direction is given by the associated
  unit vector ( ).

generally used
used in the textbook
4
Adding Vectors Using Unit Vectors

• Using
• Then
• Then Rx Ax Bx and Ry Ay By

Separately add x-components and y-components. A
sign of each component is important!
5
Subtracting Vectors Using Unit Vectors

• Using
• Then
• Then Cx Ax - Bx and Cy Ay - By

Separately subtract x-components and
y-components. A sign of each component is
important!
6
Unit Vectors, Example

• Example Find the sum and difference of two
  vectors A and B lying in the xy plane given by

7
Trig Function Warning

• The component equations (Ax A cos q and Ay A
  sin q) apply only when the angle is measured with
  respect to the x-axis (ccw from the positive
  x-axis).
• The resultant angle (tan q Ay / Ax) must be
  determined with respect to the x-axis.
• You can always think about the actual triangle
  being formed and what angle you know and apply
  the appropriate trig functions

8
Conceptual Questions (p. 72 at the end of Ch. 3)

• Q. 3. Given that (a) how
  does the magnitude of compare with the
  magnitude of (b) How does the direction of
  compare with the direction of ?
• Q. 4. Can a component of a vector be greater than
  the vectors magnitude?
• Q. 9. Suppose that and have nonzero
  magnitude. Is it possible for to be zero?
• Q. 10. Vector has x and y components of
  equal magnitude. What can you say about the
  possible directions of
• Q. 11. Given that and that
  how are and
  oriented relative to one another?

9
Problem solving

• Problem 19 p. 74. (Adding and subtracting
  vectors) Vector points in the negative y
  direction and has a magnitude of 5 units. Vector
  has twice the magnitude and points in the
  positive x direction. Find the direction and
  magnitude of (a) , (b) , and (c)
• Problem 25 p. 74. (Unit vectors) Find the
  direction and magnitude of the vectors (a)
  (b) and (c)

10
Displacement, Velocity, AccelerationWalker Ch. 3
Sect. 3-5

• Position vector
• Displacement vector
• Velocity
• Acceleration

11
Average Acceleration for a Car Traveling in a
Circle With Constant Speed Figure 3-23 p. 66 of
Walker

• Speed of the car never changes, but the car is
  accelerating due to the change in the direction
  of its motion

12
Problem solving

• Problem 33 p. 75. What is the direction and
  magnitude of your total displacement if you have
  traveled due west with a speed of 20.0 m/s for
  120 s, then due south at 15 m/s for 60.0 s?
• Problem 34 p. 75. You drive a car 1500 ft to
  the east, then 2500 ft to the north. If the trip
  took 3.0 minutes, what was the direction and
  magnitude of your average velocity?
• Problem 35 p 75. A jogger runs with a speed of
  3.25 m/s in a direction 30.0 above the x axis.
  (a) Find the x and y components of the joggers
  velocity. (b) How will the velocity components
  found in part (a) change if the joggers speed is
  halved?