APMA 2120 Final Exam Review
What is a unit vector? - correct answer ✔✔A vector that preserves the direction of the parent vector but
has magnitude of 1
1/||v|| * v
How do you parametrize a line? - correct answer ✔✔r = P + tv
P = a point
v = a vector
How do you parametrize a line using two points? - correct answer ✔✔r = (1-t) * P1 + t * P2
What is the geometric expression for the dot product? - correct answer ✔✔cos(theta) =
(v dot w) / ||v||||w||
What can you tell from a dot product's result? - correct answer ✔✔Acute if v dot w > 0
Perp. if v dot w = 0
Obtuse if v dot w < 0
How can you find the scalar component of a vector u on another vector v? - correct answer ✔✔||u||
cos(theta) = (u dot v) / ||v||
How can you find the vector projection of one vector on another? - correct answer ✔✔(u dot v) / ||v||
* v / ||v||
What is the equation of an ellipsoid? - correct answer ✔✔(x/a)^2 + (y/b)^2 + (z/c)^2 = 1
intercepts are +- a, +- b, +- c
What is the equation of a hyperboloid? - correct answer ✔✔One Sheet: (x/a)^2 + (y/b)^2 = (z/c)^2 + 1
, Two Sheets: (x/a)^2 + (y/b)^2 = (z/c)^2 - 1
What is the equation of an elliptic cone? - correct answer ✔✔(x/a)^2 + (y/b)^2 = (z/c)^2
What is the equation of a paraboloid? - correct answer ✔✔Elliptic: z = (x/a)^2 + (y/b)^2
Hyperbolic: z = (x/a)^2 - (y/b)^2
What is the equation for a tangent line at r(t) - correct answer ✔✔L(t) = r(t0) + r * r'(t0)
How do you find arc length? - correct answer ✔✔s = integral(||r'(t)||) dt from a to b
What are the steps for finding an arc-length parametrization? - correct answer ✔✔1. Start with a given
r(t)
2. Solve s(t) = integral(||r'(u)|| du from 0 to t
3. Replace t with s/r--the arc length product
4. define r(s) = r(g(s))
How do you find the Tangent Unit Vector? - correct answer ✔✔T(t) = r'(t)/ ||r'(t)||
How do you find curvature? - correct answer ✔✔k(s) = ||dT(s)||/ds = ||dr'(s) / ds|| or
k(t) = ||r'(t) x r''(t)|| / ||r'(t)||^3
How do you find the Unit Normal Vector? - correct answer ✔✔N(t) = T'(t) / ||T'(t)||
What is the breakdown for the acceleration vector? - correct answer ✔✔a(t) = aT(t) * T(t) + aN(t) * N(t)
In total, a(t) = aT(t) + aN(t)
How can you find aT(t)?
How can you find aN(t)? - correct answer ✔✔aT(t) = a(t) dot T(t)
What is a unit vector? - correct answer ✔✔A vector that preserves the direction of the parent vector but
has magnitude of 1
1/||v|| * v
How do you parametrize a line? - correct answer ✔✔r = P + tv
P = a point
v = a vector
How do you parametrize a line using two points? - correct answer ✔✔r = (1-t) * P1 + t * P2
What is the geometric expression for the dot product? - correct answer ✔✔cos(theta) =
(v dot w) / ||v||||w||
What can you tell from a dot product's result? - correct answer ✔✔Acute if v dot w > 0
Perp. if v dot w = 0
Obtuse if v dot w < 0
How can you find the scalar component of a vector u on another vector v? - correct answer ✔✔||u||
cos(theta) = (u dot v) / ||v||
How can you find the vector projection of one vector on another? - correct answer ✔✔(u dot v) / ||v||
* v / ||v||
What is the equation of an ellipsoid? - correct answer ✔✔(x/a)^2 + (y/b)^2 + (z/c)^2 = 1
intercepts are +- a, +- b, +- c
What is the equation of a hyperboloid? - correct answer ✔✔One Sheet: (x/a)^2 + (y/b)^2 = (z/c)^2 + 1
, Two Sheets: (x/a)^2 + (y/b)^2 = (z/c)^2 - 1
What is the equation of an elliptic cone? - correct answer ✔✔(x/a)^2 + (y/b)^2 = (z/c)^2
What is the equation of a paraboloid? - correct answer ✔✔Elliptic: z = (x/a)^2 + (y/b)^2
Hyperbolic: z = (x/a)^2 - (y/b)^2
What is the equation for a tangent line at r(t) - correct answer ✔✔L(t) = r(t0) + r * r'(t0)
How do you find arc length? - correct answer ✔✔s = integral(||r'(t)||) dt from a to b
What are the steps for finding an arc-length parametrization? - correct answer ✔✔1. Start with a given
r(t)
2. Solve s(t) = integral(||r'(u)|| du from 0 to t
3. Replace t with s/r--the arc length product
4. define r(s) = r(g(s))
How do you find the Tangent Unit Vector? - correct answer ✔✔T(t) = r'(t)/ ||r'(t)||
How do you find curvature? - correct answer ✔✔k(s) = ||dT(s)||/ds = ||dr'(s) / ds|| or
k(t) = ||r'(t) x r''(t)|| / ||r'(t)||^3
How do you find the Unit Normal Vector? - correct answer ✔✔N(t) = T'(t) / ||T'(t)||
What is the breakdown for the acceleration vector? - correct answer ✔✔a(t) = aT(t) * T(t) + aN(t) * N(t)
In total, a(t) = aT(t) + aN(t)
How can you find aT(t)?
How can you find aN(t)? - correct answer ✔✔aT(t) = a(t) dot T(t)
