- MLC hours when Calc 3 specialists are available
- Monday 11:30 - 3:30pm
- Tuesday 11:30 - 12:30pm, 1:30 - 3:30pm, 7:00 - 10:00pm
- Wednesday 11:30 - 2:30pm, 7:00 - 10:00pm
- Thursday 2:30 - 3:30pm, 7:00 - 10:00pm
- Friday 11:30 - 2:30pm
- Second part of section 12.5
- Equations
- Distance from point to line
- Equations for planes
- Distance from point to plane
- Angle at which planes intersect
- Intersections of planes
- Normal vector to a plane describes “tilt” of plane
- Constructing planes?
- From normal and point in plane
- But there are lots of ways to get a normal
- Where does equation for plane come from?
- Equations
- Examples
- Equation for plane through (1,0,1) perpendicular to ⟨2,1,3⟩
- Vector is normal and so gives coefficients
- 2(x-1) + y + 3(z-1) = 0
- 2x - 2 + y + 3z - 3 = 0
- 2x + y + 3z = 5
- Or 2x + y + 3z = D = 2(1) + 0 + 3(1) = 5
- Equation for plane through (1,-2,1), (0,-1,2), (2,1,1)
- Meaning of D?
- Distance from origin to plane if normal is unit vector
- How do different equations for the same plane relate to each other?
- Scalar multiples
- Meaning of D?
- Can you talk about the angle at which a plane is tilted?
- Usually it’s done in terms of angles to each axis, in the form of
“direction cosines”
- Usually it’s done in terms of angles to each axis, in the form of
“direction cosines”
- Equation for plane containing lines (1,2,3) + t⟨0,1,1⟩ and (1,3,4) + t⟨-1,2,1⟩
- Beware that lines really do intersect
- Check by checking that points from both lines satisfy plane equation
- Otherwise you can still find normal to all planes parallel to the lines, but no single plane contains both lines
- Beware that lines really do intersect
- Equation for plane through (1,0,1) perpendicular to ⟨2,1,3⟩
- Curves in space (aka vector-valued functions)
- Read section 13.1
- Bring computers for muPad