APMA 2120 Final Question and
answers verified to pass 2025/2026
Where do saddle points exist on a contour map? - correct answer ✔Where the lines intersect
Distance formula - correct answer ✔sqrt[(x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2]
Dot product formula (a dot b) - correct answer ✔|a||b|cos(theta)
Two vectors are orthogonal if - correct answer ✔Their dot product = 0
Two vectors are parallel if - correct answer ✔a x b = 0
Scalar projection of a onto b - correct answer ✔(a dot b) / |b|
Vector projection of a onto b - correct answer ✔[(a dot b) / |b|^2] * b
Volume of a parallelepiped - correct answer ✔|a dot (b x c)|
Scalar equation of a plane - correct answer ✔a(x-x0)+b(y-y0)+c(z-z0)=0
Linear equation of a plane - correct answer ✔ax + by + cz + d = 0
Vector equation of a plane - correct answer ✔n • (r - r0) = 0
, Parametric equations of a line - correct answer ✔x = x0 + at
y = y0 + bt
z = z0 + ct
Symmetric equations of a line - correct answer ✔(x-x0)/a = (y-y0)/b = (z-z0)/c
Area of a parallelogram - correct answer ✔|a x b|
Area of a triangle - correct answer ✔1/2|a x b|
Unit vector of v - correct answer ✔v/|v|
How can you tell if r(t) is continuous at t = a? - correct answer ✔if lim t->a r(t) = r(a)
Unit tangent vector - correct answer ✔T(t) = r'(t) / |r'(t)|
Arclength - correct answer ✔
Unit normal vector - correct answer ✔N(t) = T'(t)/|T'(t)|
Unit binormal vector - correct answer ✔B(t) = T(t) x N(t)
What is the Jacobian change of variables for cylindrical coordinates? - correct answer ✔rdrd(theta)dz
What is the Jacobian change of variables for spherical coordinate? - correct answer ✔rho^2 * sin(phi)
d(rho) d(theta) d(phi)
answers verified to pass 2025/2026
Where do saddle points exist on a contour map? - correct answer ✔Where the lines intersect
Distance formula - correct answer ✔sqrt[(x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2]
Dot product formula (a dot b) - correct answer ✔|a||b|cos(theta)
Two vectors are orthogonal if - correct answer ✔Their dot product = 0
Two vectors are parallel if - correct answer ✔a x b = 0
Scalar projection of a onto b - correct answer ✔(a dot b) / |b|
Vector projection of a onto b - correct answer ✔[(a dot b) / |b|^2] * b
Volume of a parallelepiped - correct answer ✔|a dot (b x c)|
Scalar equation of a plane - correct answer ✔a(x-x0)+b(y-y0)+c(z-z0)=0
Linear equation of a plane - correct answer ✔ax + by + cz + d = 0
Vector equation of a plane - correct answer ✔n • (r - r0) = 0
, Parametric equations of a line - correct answer ✔x = x0 + at
y = y0 + bt
z = z0 + ct
Symmetric equations of a line - correct answer ✔(x-x0)/a = (y-y0)/b = (z-z0)/c
Area of a parallelogram - correct answer ✔|a x b|
Area of a triangle - correct answer ✔1/2|a x b|
Unit vector of v - correct answer ✔v/|v|
How can you tell if r(t) is continuous at t = a? - correct answer ✔if lim t->a r(t) = r(a)
Unit tangent vector - correct answer ✔T(t) = r'(t) / |r'(t)|
Arclength - correct answer ✔
Unit normal vector - correct answer ✔N(t) = T'(t)/|T'(t)|
Unit binormal vector - correct answer ✔B(t) = T(t) x N(t)
What is the Jacobian change of variables for cylindrical coordinates? - correct answer ✔rdrd(theta)dz
What is the Jacobian change of variables for spherical coordinate? - correct answer ✔rho^2 * sin(phi)
d(rho) d(theta) d(phi)
