Midterm Review β October 11, 2013
Topics: Everything
Vectors & Geometry
Dot Product πβ β π = β π π = π |β οΏ½ποΏ½cosπ
π π
Finding angles, components, detecting β₯
Determinants (2d, 3d= π β β (π Γ β))
οΏ½
π€Μ π₯Μ π
Cross Product πβ Γ π = οΏ½ π1 π 2 π 3οΏ½
π1 π2 π 3
Direction β₯ π β, π
Length= Area of parallelogram formed byπ β, π
Equations of planes ππ₯ + ππ¦ + ππ§ = π, < π,π,π >= normal vector
Equations of lines
π₯ = π₯ 0 ππ‘
Parametric equationοΏ½ π¦ = π¦ 0 + ππ‘ , < π,π,π > β₯ to the line
π§ = π§ + ππ‘ < π₯ 0π¦ 0π§ >0is point on line
0
(Symmetric equations)
Geometry of lines and planes
(Intersections, angles, distances(see Figure 1))
Ex. Plane through A (0, 1, 2), B (2, -1, 3), C (1, 0, 1)
π΄π΅ =< 2,β2,1 > π΄πΆ =< 1,β1,β1 >
π€Μ π₯Μ π
οΏ½β = π΄π΅ Γ π΄πΆ = οΏ½β οΏ½ =< 2 + 1,β β2 β 1 ,β2 + 2 >=< 3,3,0 >
2 β2 1
1 β1 β1
3π₯ + 3π¦ + 0π§ = 3 0 + 3 1 + 0 2 = 3 or π₯ + π¦ = 1
Parametric curves (split between Chapters 10 & 13)
Finding parametric equations geometrically
Velocityπ£ β = ππ, Acceleration πβ = ππ£β
ππ‘ ππ‘
Speed = π£|β , Unit Tangent π = π£β
π£β|
Arc Length π

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