Multivariable Calculus – Questions With Correct
Answers
Vector parameterization
Given a position vector (direction) or P and a point, find the equation of the
line passing through that point, parallel to position vector. ✔️Ans - r(t) =
r₀ + tv = <x₀, y₀, z₀> + t <a, b, c>
r(t) describes a vector whose terminal point traces out a line from -∞ to ∞
Parametric equations ✔️Ans - x = x₀ + at
y = y₀ + at
z = z₀ + at
Do these equations parameterize the same line?
r1(t) = (3, −1, 4) + t<8, 12, −6>
r2(t) = (11, 11, −2) + t<4, 6, −3> ✔️Ans - Two lines in R3 coincide (are the
same) if they are parallel and pass through a common point.
thus, v = λ(w)
and same t must work in parametric equations
Sphere of radius R and center (a,b,c) ✔️Ans - (x-a)² + (y-b)² + (z-c)² = R²
Cylinder of radius R with vertical axis (a,b,0) ✔️Ans - (x-a)² + (y-b)² = R²
Length/magnitude of v = ‖PQ‖ ✔️Ans - √(a₂-a₁)² + (b₂-b₁)² + (c₂-c₁)²
How many different direction vectors does a line have? ✔️Ans - Infinitely
many direction vectors
Multiply the vectors with ✔️Ans - the dot product
v·w=w·v? ✔️Ans - true
Second, the dot product of a vector with itself is:
v · v: ✔️Ans - ‖v‖²
, Pulling out scalars with dot product ✔️Ans - (λv) · w = v · (λw) = λ(v · w)
Distributive law with dot product ✔️Ans - u · (v + w) = u · v + u · w
Prove :
v · w = ‖v‖ ‖w‖ cosθ ✔️Ans - Cosines: ‖v-w‖² = ‖v‖² + ‖w‖² -
2cosθ‖v‖‖w‖
‖v-w‖² = (v · v) - 2(v · w) + (w · w) = ‖v‖² + ‖w‖² - 2(v · w)
‖v‖² + ‖w‖² - 2(v · w) = ‖v‖² + ‖w‖² - 2cosθ ‖v‖ ‖w‖
- 2(v · w) = - 2cosθ ‖v‖ ‖w‖
v · w = cosθ ‖v‖ ‖w‖
law of cosines
‖v-w‖² = ✔️Ans - ‖v‖² + ‖w‖² - 2cosθ‖v‖‖w‖
Two nonzero vectors v and w are called perpendicular or orthogonal if the
angle between them is ✔️Ans - π/2
v · w = ‖v‖ ‖w‖ cosθ = 0 ✔️Ans - v · w = 0
, the orthogonal
because i =〈1, 0, 0〉, j =〈0, 1, 0〉, and k =〈0, 0, 1 〉 ✔️Ans - i · j = i · k = j · k
and
i·i=j·j=k·k
v·w= ✔️Ans - ‖v‖ ‖w‖ cosθ
projection of u onto v ✔️Ans - U‖v = ( u · v / ‖v‖²) * v
or
(u·v/v·v)v
component of u along v ✔️Ans - u · v / ‖v‖
Area of a parallelogram spanned by v and w ✔️Ans - ‖ v x w ‖
v x w using geometric properties ✔️Ans - ‖ v x w ‖ = ‖ v ‖‖ w ‖ sinθ
Answers
Vector parameterization
Given a position vector (direction) or P and a point, find the equation of the
line passing through that point, parallel to position vector. ✔️Ans - r(t) =
r₀ + tv = <x₀, y₀, z₀> + t <a, b, c>
r(t) describes a vector whose terminal point traces out a line from -∞ to ∞
Parametric equations ✔️Ans - x = x₀ + at
y = y₀ + at
z = z₀ + at
Do these equations parameterize the same line?
r1(t) = (3, −1, 4) + t<8, 12, −6>
r2(t) = (11, 11, −2) + t<4, 6, −3> ✔️Ans - Two lines in R3 coincide (are the
same) if they are parallel and pass through a common point.
thus, v = λ(w)
and same t must work in parametric equations
Sphere of radius R and center (a,b,c) ✔️Ans - (x-a)² + (y-b)² + (z-c)² = R²
Cylinder of radius R with vertical axis (a,b,0) ✔️Ans - (x-a)² + (y-b)² = R²
Length/magnitude of v = ‖PQ‖ ✔️Ans - √(a₂-a₁)² + (b₂-b₁)² + (c₂-c₁)²
How many different direction vectors does a line have? ✔️Ans - Infinitely
many direction vectors
Multiply the vectors with ✔️Ans - the dot product
v·w=w·v? ✔️Ans - true
Second, the dot product of a vector with itself is:
v · v: ✔️Ans - ‖v‖²
, Pulling out scalars with dot product ✔️Ans - (λv) · w = v · (λw) = λ(v · w)
Distributive law with dot product ✔️Ans - u · (v + w) = u · v + u · w
Prove :
v · w = ‖v‖ ‖w‖ cosθ ✔️Ans - Cosines: ‖v-w‖² = ‖v‖² + ‖w‖² -
2cosθ‖v‖‖w‖
‖v-w‖² = (v · v) - 2(v · w) + (w · w) = ‖v‖² + ‖w‖² - 2(v · w)
‖v‖² + ‖w‖² - 2(v · w) = ‖v‖² + ‖w‖² - 2cosθ ‖v‖ ‖w‖
- 2(v · w) = - 2cosθ ‖v‖ ‖w‖
v · w = cosθ ‖v‖ ‖w‖
law of cosines
‖v-w‖² = ✔️Ans - ‖v‖² + ‖w‖² - 2cosθ‖v‖‖w‖
Two nonzero vectors v and w are called perpendicular or orthogonal if the
angle between them is ✔️Ans - π/2
v · w = ‖v‖ ‖w‖ cosθ = 0 ✔️Ans - v · w = 0
, the orthogonal
because i =〈1, 0, 0〉, j =〈0, 1, 0〉, and k =〈0, 0, 1 〉 ✔️Ans - i · j = i · k = j · k
and
i·i=j·j=k·k
v·w= ✔️Ans - ‖v‖ ‖w‖ cosθ
projection of u onto v ✔️Ans - U‖v = ( u · v / ‖v‖²) * v
or
(u·v/v·v)v
component of u along v ✔️Ans - u · v / ‖v‖
Area of a parallelogram spanned by v and w ✔️Ans - ‖ v x w ‖
v x w using geometric properties ✔️Ans - ‖ v x w ‖ = ‖ v ‖‖ w ‖ sinθ
