- Standard position?
- Makes one of the endpoints of a vector (0,0,0) so you don’t have to think as much about the vector being a difference of points
- Unit vector?
- Vector of length 1, provides a standardized measure of distance in a direction
- One role for unit vectors: consider traditional derivative in 2 dimensions as
slope, i.e., rise per unit run
- In more than 2 dimensions, that “run” could be measured in any direction
- Thinking of slope as rise per unit vector is helpful
- Vector of length 1, provides a standardized measure of distance in a direction
- Section 12.2
- Force, displacement, velocity
- Force has direction and magnitude
- Vector points in force’s direction, length = strength
- Displacement has distance and direction
- Velocity has direction and speed
- Force has direction and magnitude
- Vector AB starts at A, goes to B, has magnitude |AB|?
- |AB| = length of vector aka distance from A to B
- Also distance from origin to endpoint of standard vector
- Notation corresponds to absolute value notation as distance from zero point
on a number line
- 2D vector v in plane from (0,0) to (v1,v2) has
component form ⟨v1,v2⟩,
⟨v1,v2,v3⟩ in 3D…?
- Force, displacement, velocity
- Applications
- Directions in room
- To walk from origin to (9,4,0), walk along vector 9i + 4j = ⟨9,4,0⟩
- To fly from (1,2,1) to (10,6,1), also follow vector 9i + 4j = ⟨9,4,0⟩
- Zigzag path
- Go from (0,0,0) along ⟨3,2,0⟩ then ⟨1,3,2⟩, then ⟨-2,4,2⟩
- End point:
- Origin + ⟨3,2,0⟩ + ⟨1,3,2⟩ + ⟨-2,4,2⟩ = origin + ⟨2,9,4⟩ = (2,9,4)
- Vector addition
- Directions in room
- Further examples and discussion of vectors