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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Find the intersection of the line and the plane.x − z = 6, r(t) = (1, 0, −1) + t (4, 9, 2)
Find the trace of the plane in the given coordinate plane.3x − 9y + 4z = 5, yz
Find the trace of the plane in the given coordinate plane.3x − 9y + 4z = 5, xz
Find the trace of the plane in the given coordinate plane.3x + 4z = −2, xy
Find the trace of the plane in the given coordinate plane.3x + 4z = −2, xz
Find the trace of the plane in the given coordinate plane.−x + y = 4, xz
Find the trace of the plane in the given coordinate plane.−x + y = 4, yz
Does the plane x = 5 have a trace in the yz-plane? Explain.
Give equations for two distinct planes whose trace in the xy-plane has equation 4x + 3y = 8.
Give equations for two distinct planes whose trace in the yz-plane has equation y = 4z.
Find parametric equations for the line through P0 = (3, −1, 1) perpendicular to the plane 3x + 5y − 7z = 29.
Find all planes in R3 whose intersection with the xz-plane is the line with equation 3x + 2z = 5.
Find all planes in R3 whose intersection with the xy-plane is the line r(t) = t(2, 1, 0).
Compute the angle between the two planes, defined as the angle θ (between 0 and π) between their normal vectors (Figure 10).Planes with normals n1 = (1, 2, 1), n2 = (4, 1, 3) n 112 n2 0 n PL P 0 L
Compute the angle between the two planes, defined as the angle θ (between 0 and π) between their normal vectors (Figure 10).2x + 3y + 7z = 2 and 4x − 2y + 2z = 4 n 112 n2 0 n PL P 0 L
Compute the angle between the two planes, defined as the angle θ (between 0 and π) between their normal vectors (Figure 10).3(x − 1) − 5y + 2(z − 12) = 0 and the plane with normal n = (1, 0,
Compute the angle between the two planes, defined as the angle θ (between 0 and π) between their normal vectors (Figure 10).x − 3y + z = 3 and 2x − 3z = 4 n 112 n2 0 n PL P 0 L
Compute the angle between the two planes, defined as the angle θ (between 0 and π) between their normal vectors (Figure 10).The plane through (1, 0, 0), (0, 1, 0), and (0, 0, 1) and the yz-plane n
Find an equation of a plane making an angle of π/2 with the plane 3x + y − 4z = 2.
Let P1 and P2 be planes with normal vectors n1 and n2. Assume that the planes are not parallel, and let L be their intersection (a line). Show that n1 × n2 is a direction vector for L.
Find a plane that is perpendicular to the two planes x + y = 3 and x + 2y − z = 4.
Let L be the intersection of the planes x + y + z = 1 and x + 2y + 3z = 1. Use Exercise 64 to find a direction vector for L. Then find a point P on L by inspection, and write down the parametric
Let L denote the intersection of the planes x − y − z = 1 and 2x + 3y + z = 2. Find parametric equations for the line L. To find a point on L, substitute an arbitrary value for z (say, z = 2) and
Find parametric equations for the intersection of the planes 2x + y − 3z = 0 and x + y = 1.
The planeintersects the x-, y-, and z-axes in points P, Q, and R. Find the area of the triangle ΔPQR. || NIM + + XIN
In this exercise, we show that the orthogonal distance D from the plane P with equation ax + by + cz = d to the origin O is equal to (Figure 11) D = |d| a + b + c
Two vectors v and w, each of length 12, lie in the plane x + 2y − 2z = 0. The angle between v and w is π/6. This information determines v × w up to a sign ±1. What are the two possible values of
Let P be a plane with equation ax + by + cz = d and normal vector n = (a, b, c). For any point Q, there is a unique point P on P that is closest to Q, and is such that PQ is orthogonal to P(Figure
By definition, the distance from Q = (x1, y1, z1) to the plane P is the distance to the point P on P closest to Q. Prove distance from Q to P= lax + by + cz - dl ||n||
Use Eq. (5) to find the point P nearest to Q = (2, 1, 2) on the plane x + y + z = 1. OP=00+ d-00 n n.n n
Use Eq. (6) to find the distance from Q = (1, 1, 1) to the plane 2x + y + 5z = 2. distance from Q to P = lax + by + cz - d| ||n||
Find the point P nearest to Q = (−1, 3, −1) on the plane x − 4z = 2.
Find the distance from Q = (1, 2, 2) to the plane n · (x, y, z) = 3, where n = (3/5,, 4/5, 0).
What is the distance from Q = (a, b, c) to the plane x = 0? Visualize your answer geometrically and explain without computation. Then verify that Eq. (6) yields the same answer.
The equation of a plane n · (x, y, z) = d is said to be in normal form if n is a unit vector. Show that in this case, |d| is the distance from the plane to the origin. Write the equation of the
Convert from cylindrical to rectangular coordinates. 2, -8) 3'
Convert from cylindrical to rectangular coordinates.(4, π, 4)
Convert from cylindrical to rectangular coordinates. klin 5' 2,
Convert from cylindrical to rectangular coordinates. (1,77,-2)
Convert from rectangular to cylindrical coordinates.(1, −1, 1)
Convert from rectangular to cylindrical coordinates.(2, 2, 1)
Convert from rectangular to cylindrical coordinates. 3 3V3 2 2
Convert from rectangular to cylindrical coordinates.(1,√3, 7)
Convert from rectangular to cylindrical coordinates. 5 5 2 2 ,2
Convert from rectangular to cylindrical coordinates.(3, 3 √3, 2)
Describe the set in cylindrical coordinates.x2 + y2 ≤ 3
Describe the set in cylindrical coordinates.x2 + y2 + z2 ≤ 10
Describe the set in cylindrical coordinates.y2 + z2 ≤ 4, x = 0
Describe the set in cylindrical coordinates.x2 + y2 + z2 = 9, y ≥ 0, z ≥ 0
Describe the set in cylindrical coordinates.x2 + y2 ≤ 9, x ≥ y
Describe the set in cylindrical coordinates.y2 + z2 ≤ 9, x ≥ y
Sketch the set (described in cylindrical coordinates).r = 4
Sketch the set (described in cylindrical coordinates).θ = π/3
Sketch the set (described in cylindrical coordinates).z = −2
Sketch the set (described in cylindrical coordinates).r = 2, z = 3
Sketch the set (described in cylindrical coordinates).1 ≤ r ≤ 3, 0 ≤ z ≤ 4
Sketch the set (described in cylindrical coordinates).z = r
Sketch the set (described in cylindrical coordinates).r = sin θ
Sketch the set (described in cylindrical coordinates).1 ≤ r ≤ 3, 0 ≤ θ ≤ π/2, 0≤ z ≤ 4
Sketch the set (described in cylindrical coordinates).z2 + r2 ≤ 4
Sketch the set (described in cylindrical coordinates).r ≤ 3, π ≤ θ ≤ 3π/2, z = 4
Find an equation of the form r = ƒ(θ, z) in cylindrical coordinates for the following surfaces.z = x + y
Find an equation of the form r = ƒ(θ, z) in cylindrical coordinates for the following surfaces.x2 + y2 + z2 = 2
Find an equation of the form r = ƒ(θ, z) in cylindrical coordinates for the following surfaces.x2/yz = 1
Find an equation of the form r = ƒ(θ, z) in cylindrical coordinates for the following surfaces.x2 − y2 = 4
Find an equation of the form r = ƒ(θ, z) in cylindrical coordinates for the following surfaces.x2 + y2 = 4
Convert from spherical to rectangular coordinates. (3,0,
Convert from spherical to rectangular coordinates. -3 -4
Find an equation of the form r = ƒ(θ, z) in cylindrical coordinates for the following surfaces.z = 3xy
Convert from spherical to rectangular coordinates. 3 4
Convert from spherical to rectangular coordinates.(3, π, 0)
Convert from spherical to rectangular coordinates. 6, 5 6' 6
Convert from spherical to rectangular coordinates.(0.5, 3.7, 2)
Convert from rectangular to spherical coordinates. V3 3 2 2
Convert from rectangular to spherical coordinates.(√3, 0, 1)
Convert from rectangular to spherical coordinates.(1, 1, 1)
Convert from rectangular to spherical coordinates. 2 V3 2
Convert from rectangular to spherical coordinates.(1, −1, 1)
Convert from rectangular to spherical coordinates. 2 2 2 2
Convert from cylindrical to spherical coordinates.(2, 0, 2)
Convert from cylindrical to spherical coordinates.(3, π,√3)
Convert from spherical to cylindrical coordinates.(4, 0, π/4)
Convert from spherical to cylindrical coordinates.(2, π/3, π/6)
Describe the given set in spherical coordinates.x2 + y2 + z2 ≤ 100
Describe the given set in spherical coordinates.x2 + y2 + z2 = 1, z ≥ 0
Describe the given set in spherical coordinates.x2 + y2 + z2 = 10, x ≥ 0, y ≥ 0, z ≥ 0
Describe the given set in spherical coordinates.x2 + y2 + z2 ≤ 1, x = y, x ≥ 0, y ≥ 0
Describe the given set in spherical coordinates.y2 + z2 ≤ 4, x = 0
Describe the given set in spherical coordinates.x2 + y2 = 3z2
Sketch the set of points (described in spherical coordinates).ρ = 4
Sketch the set of points (described in spherical coordinates).ϕ = π/4
Sketch the set of points (described in spherical coordinates).ρ = 2, θ = π/4
Sketch the set of points (described in spherical coordinates).ρ = 2, ϕ = π/4
Sketch the set of points (described in spherical coordinates). 0 = 2 = 4 1
Sketch the set of points (described in spherical coordinates).ρ = 2, 0 ≤ ϕ ≤ π/2
Sketch the set of points (described in spherical coordinates). 0522 2 OST P2, 0 0
Sketch the set of points (described in spherical coordinates). p = 1, 3 < < 27 3
Sketch the set of points (described in spherical coordinates).ρ = csc ϕ
Sketch the set of points (described in spherical coordinates).ρ = csc ϕ cot ϕ
Find an equation of the form ρ = ƒ(θ, φ) in spherical coordinates for the following surfaces.x2 + y2 = 9

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