Question 1

Find the point at which the two lines $\mathbf { r } = \langle 1,1,0 \} + t \langle 1 , - 1,2 \}$ and $\mathbf { r } = \langle 2,0,2 \} + s ( - 1,1,0 \}$ intersect.

Accepted Answer

A) (1, 0, 1) 
B) (2, 0, 2) 
C) (1, 1, 1) 
D) (2, 2, 2) 
E) (1, 1, 0)
F)(2, 2, 0)
G)(0, 1, 1)
H)(0, 2, 2) 
A) (1, 0, 1) 
B) (2, 0, 2) 
C) (1, 1, 1) 
D) (2, 2, 2) 
E) (1, 1, 0)
F)(2, 2, 0)
G)(0, 1, 1)
H)(0, 2, 2)

Question 2

Identify the trace of the surface $x = 2 y ^ { 2 } + 3 z ^ { 2 }$ in the plane x = 1.

Accepted Answer

A) ellipse but not a circle 
B) parabola 
C) hyperbola 
D) circle 
E) two parallel straight lines
F)two intersecting straight lines
G)point
H)straight line 
A) ellipse but not a circle 
B) parabola 
C) hyperbola 
D) circle 
E) two parallel straight lines
F)two intersecting straight lines
G)point
H)straight line

Question 3

Find the domain of the function f(x, y) = $\ln \left( x - y ^ { 2 } \right)$ .

Accepted Answer

A) All points on or to the left of $x = y ^ { 2 }$ e.All points on or to the left of x = 0 
B) All points on or to the right of $x = y ^ { 2 }$ f.All points on or to the right of x = 0 
C) All points to the left of $x = y ^ { 2 }$ g.All points to the left of x = 0 
D) All points to the right of $x = y ^ { 2 }$ h.All points in the xy-plane 
A) All points on or to the left of $x = y ^ { 2 }$ e.All points on or to the left of x = 0 
B) All points on or to the right of $x = y ^ { 2 }$ f.All points on or to the right of x = 0 
C) All points to the left of $x = y ^ { 2 }$ g.All points to the left of x = 0 
D) All points to the right of $x = y ^ { 2 }$ h.All points in the xy-plane

Question 4

Given two distinct nonzero vectors a and b, is it always true that $\left| \operatorname { proj } _ { \mathbf { a } } \mathbf { b } \right| = \left| \operatorname { proj } _ { \mathbf { b } } \mathbf { a } \right|$ ?

Accepted Answer

The answer of Given two distinct nonzero vectors a and...

Question 5

Identify the surface $x = y ^ { 2 } + 2 z ^ { 2 }$ .

Accepted Answer

A) ellipsoid but not a sphere 
B) hyperboloid of one sheet 
C) hyperboloid of two sheets 
D) cylinder 
E) sphere
F)elliptic but not circular paraboloid
G)cone
H)circular paraboloid (figure of revolution) 
A) ellipsoid but not a sphere 
B) hyperboloid of one sheet 
C) hyperboloid of two sheets 
D) cylinder 
E) sphere
F)elliptic but not circular paraboloid
G)cone
H)circular paraboloid (figure of revolution)

Question 6

Find the distance between the sphere $( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + z ^ { 2 } = \frac { 1 } { 4 }$ and the sphere $( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 1$

Accepted Answer

The closest distance between t

Question 7

Let f(x, y) = $\sqrt { 4 - x ^ { 2 } - 2 y ^ { 2 } }$ (a) Evaluate $f ( - 1 , - 1 )$ .(b) Find the domain of f.(c) Find the range of f.

Accepted Answer

The answer of Let f(x, y) = \(\sqrt { 4...

Question 8

Find a unit vector orthogonal to the plane through the points (1, 0, 0), (0, 1, 0), and (0, 2, 2).

Accepted Answer

A)  $\left( \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } \right)$ 
B)  $\left( - \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } \right)$ 
C)  $\left( \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , 0 \right)$ 
D)  $\left( - \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , 0 \right)$ 
E)  $\left\langle \frac { 2 } { 3 } , \frac { 2 } { 3 } , \frac { 1 } { 3 } \right\}$ 
F) $\left( - \frac { 2 } { 3 } , \frac { 2 } { 3 } , \frac { 1 } { 3 } \right)$ 
G) $\left( \frac { 2 } { 3 } , - \frac { 2 } { 3 } , \frac { 1 } { 3 } \right)$ 
H) $\left( \frac { 2 } { 3 } , \frac { 2 } { 3 } , - \frac { 1 } { 3 } \right)$ 
A)  $\left( \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } \right)$ 
B)  $\left( - \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } \right)$ 
C)  $\left( \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , 0 \right)$ 
D)  $\left( - \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , 0 \right)$ 
E)  $\left\langle \frac { 2 } { 3 } , \frac { 2 } { 3 } , \frac { 1 } { 3 } \right\}$ 
F) $\left( - \frac { 2 } { 3 } , \frac { 2 } { 3 } , \frac { 1 } { 3 } \right)$ 
G) $\left( \frac { 2 } { 3 } , - \frac { 2 } { 3 } , \frac { 1 } { 3 } \right)$ 
H) $\left( \frac { 2 } { 3 } , \frac { 2 } { 3 } , - \frac { 1 } { 3 } \right)$

Question 9

Let l and $l ^ { \prime }$ be two lines in space given by the equations
x = 3 + t
x = -1 + t
l:
y = 1 - t $l ^ { \prime }$ :
y = 2t
z = 2t
z = 1 + kt
Find all values of k (if any) for which l and $l ^ { \prime }$ are parallel.

Accepted Answer

No value o

Find the point at which the two lines $\mathbf { r } = \langle 1,1,0 \} + t \langle 1 , - 1,2 \}$ and $\mathbf { r } = \langle 2,0,2 \} + s ( - 1,1,0 \}$ intersect.

Identify the trace of the surface $x = 2 y ^ { 2 } + 3 z ^ { 2 }$ in the plane x = 1.

Find the domain of the function f(x, y) = $\ln \left( x - y ^ { 2 } \right)$ .

Given two distinct nonzero vectors a and b, is it always true that $\left| \operatorname { proj } _ { \mathbf { a } } \mathbf { b } \right| = \left| \operatorname { proj } _ { \mathbf { b } } \mathbf { a } \right|$ ?

Identify the surface $x = y ^ { 2 } + 2 z ^ { 2 }$ .

Find the distance between the sphere $( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + z ^ { 2 } = \frac { 1 } { 4 }$ and the sphere $( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 1$

Let f(x, y) = $\sqrt { 4 - x ^ { 2 } - 2 y ^ { 2 } }$ (a) Evaluate $f ( - 1 , - 1 )$ .(b) Find the domain of f.(c) Find the range of f.

Find a unit vector orthogonal to the plane through the points (1, 0, 0), (0, 1, 0), and (0, 2, 2).

Let l and $l ^ { \prime }$ be two lines in space given by the equations x = 3 + t x = -1 + t l: y = 1 - t $l ^ { \prime }$ : y = 2t z = 2t z = 1 + kt Find all values of k (if any) for which l and $l ^ { \prime }$ are parallel.

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

Exam 9: Vectors and the Geometry of Space

Find the point at which the two lines r=⟨1,1,0}+t⟨1,−1,2}\mathbf { r } = \langle 1,1,0 \} + t \langle 1 , - 1,2 \}r=⟨1,1,0}+t⟨1,−1,2} and r=⟨2,0,2}+s(−1,1,0}\mathbf { r } = \langle 2,0,2 \} + s ( - 1,1,0 \}r=⟨2,0,2}+s(−1,1,0} intersect.

Identify the trace of the surface x=2y2+3z2x = 2 y ^ { 2 } + 3 z ^ { 2 }x=2y2+3z2 in the plane x = 1.

Find the domain of the function f(x, y) = ln⁡(x−y2)\ln \left( x - y ^ { 2 } \right)ln(x−y2) .

Given two distinct nonzero vectors a and b, is it always true that ∣proj⁡ab∣=∣proj⁡ba∣\left| \operatorname { proj } _ { \mathbf { a } } \mathbf { b } \right| = \left| \operatorname { proj } _ { \mathbf { b } } \mathbf { a } \right|∣proja​b∣=∣projb​a∣ ?

Identify the surface x=y2+2z2x = y ^ { 2 } + 2 z ^ { 2 }x=y2+2z2 .

Find the distance between the sphere (x−1)2+(y+1)2+z2=14( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + z ^ { 2 } = \frac { 1 } { 4 }(x−1)2+(y+1)2+z2=41​ and the sphere (x−3)2+(y+2)2+(z+2)2=1( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 1(x−3)2+(y+2)2+(z+2)2=1

Let f(x, y) = 4−x2−2y2\sqrt { 4 - x ^ { 2 } - 2 y ^ { 2 } }4−x2−2y2​ (a) Evaluate f(−1,−1)f ( - 1 , - 1 )f(−1,−1) .(b) Find the domain of f.(c) Find the range of f.

Find a unit vector orthogonal to the plane through the points (1, 0, 0), (0, 1, 0), and (0, 2, 2).

Let l and l′l ^ { \prime }l′ be two lines in space given by the equations x = 3 + t x = -1 + t l: y = 1 - t l′l ^ { \prime }l′ : y = 2t z = 2t z = 1 + kt Find all values of k (if any) for which l and l′l ^ { \prime }l′ are parallel.

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

Find the point at which the two lines $\mathbf { r } = \langle 1,1,0 \} + t \langle 1 , - 1,2 \}$ and $\mathbf { r } = \langle 2,0,2 \} + s ( - 1,1,0 \}$ intersect.

Identify the trace of the surface $x = 2 y ^ { 2 } + 3 z ^ { 2 }$ in the plane x = 1.

Find the domain of the function f(x, y) = $\ln \left( x - y ^ { 2 } \right)$ .

Given two distinct nonzero vectors a and b, is it always true that $\left| \operatorname { proj } _ { \mathbf { a } } \mathbf { b } \right| = \left| \operatorname { proj } _ { \mathbf { b } } \mathbf { a } \right|$ ?

Identify the surface $x = y ^ { 2 } + 2 z ^ { 2 }$ .

Find the distance between the sphere $( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + z ^ { 2 } = \frac { 1 } { 4 }$ and the sphere $( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 1$

Let f(x, y) = $\sqrt { 4 - x ^ { 2 } - 2 y ^ { 2 } }$ (a) Evaluate $f ( - 1 , - 1 )$ .(b) Find the domain of f.(c) Find the range of f.

Let l and $l ^ { \prime }$ be two lines in space given by the equations x = 3 + t x = -1 + t l: y = 1 - t $l ^ { \prime }$ : y = 2t z = 2t z = 1 + kt Find all values of k (if any) for which l and $l ^ { \prime }$ are parallel.