Overview
This lecture covers parametric equations of lines in calculus, focusing on how to express lines using vectors and parameters, and relates parametric forms to traditional line equations.
Parametric Equations of Lines
- Lines can be modeled as motion from a starting point ( p_0 ) to a point ( p_t ) over time ( t ).
- Velocity is a vector quantity (has magnitude and direction) and is used to describe motion along a line.
- The general vector form: ( p_t = p_0 + t\vec{v} ), where ( \vec{v} ) is a direction (velocity) vector.
- Parametric equations for 3D lines:
- ( x(t) = x_0 + ta )
- ( y(t) = y_0 + tb )
- ( z(t) = z_0 + tc )
- Any vector parallel to the line can be used for ( \vec{v} ).
Example: Line Through a Point Parallel to a Vector
- Given a vector ( \langle 5, -4, 8 \rangle ) and passing through ( (-2, 4, 0) ):
- ( x(t) = -2 + 5t )
- ( y(t) = 4 - 4t )
- ( z(t) = 0 + 8t )
- Plugging in ( t = 0 ) recovers the initial point; increasing ( t ) gives more points on the line.
Example: Parametric Equations from Two Points
- Given ( P_1 = (3, -7) ) and ( P_2 = (-2, -1) ):
- Find direction vector: ( (-2-3, -1-(-7)) = (-5, 6) )
- Using ( P_1 ) as start:
- ( x(t) = 3 - 5t )
- ( y(t) = -7 + 6t )
- Using ( P_2 ) as start:
- ( x(t) = -2 - 5t )
- ( y(t) = -1 + 6t )
- Plug in ( t = 0 ) to get the starting point; ( t = 1 ) yields the other point.
Converting Parametric to Slope-Intercept Form
- Solve the parametric ( x(t) ) for ( t ), then substitute into ( y(t) ).
- Example: With ( x(t) = -2 - 5t ), ( t = -\frac{1}{5}(x + 2) ).
- Substitute into ( y(t) ) and simplify to obtain ( y = mx + b ).
Matching Slope-Intercept Form via Algebra
- Compute slope ( m = \frac{\Delta y}{\Delta x} ).
- Use point-slope form to derive ( y = mx + b ).
- Both parametric conversion and traditional algebra give the same result.
Key Terms & Definitions
- Parametric Equation — An equation expressing variables as functions of a parameter, usually time ( t ).
- Vector — A mathematical object with both magnitude and direction.
- Velocity — A vector representing the rate and direction of motion.
- Direction Vector — The vector that gives the direction of the line.
- Slope-Intercept Form — The line equation ( y = mx + b ).
Action Items / Next Steps
- Practice writing parametric equations for lines given points and direction vectors.
- Convert parametric equations to slope-intercept form for comparison.
- Complete any assigned problems from section 11.5 in the Anton calculus book.