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mathematics
precalculus

Questions and Answers of Precalculus

Describe and sketch the surface.x2 + z2 = 1
(a) Sketch the graph of y = ex as a curve in R2.(b) Sketch the graph of y = ex as a surface in R3.(c) Describe and sketch the surface z = ey.
(a) What does the equation y = x2 represent as a curve in R2?(b) What does it represent as a surface in R3?(c) What does the equation z = y2 represent?
If a, b, and c are not all 0, show that the equation ax + by + cz + d = 0 represents a plane and (a, b, c) is a normal vector to the plane. Suppose a ≠ 0 and rewrite the equation in the form + b(y
Give a geometric description of each family of planes.(a) x + y + z = c (b) x + y + cz = 1(c) y cos θ + z sin θ = 1
Two tanks are participating in a battle simulation. Tank A is at point (325, 810, 561) and tank B is positioned at point (765, 675, 599).(a) Find parametric equations for the line of sight between
Let L1 be the line through the points (1, 2, 6) and (2, 4, 8). Let L2 be the line of intersection of the planes P1 and P2, where P1 is the plane x - y + 2z + 1 = 0 and P2 is the plane through the
Let L1 be the line through the origin and the point (2, 0, -1). Let L2 be the line through the points (1, -1, 1) and (4, 1, 3). Find the distance between L1 and L2.
Find the distance between the skew lines with parametric equations x = 1 + t, y = 1 + 6t, z = 2t, and x = 1 + 2s, y = 5 + 15s, z = -2 + 6s.
Show that the lines with symmetric equations x = y = z and x + 1 = y/2 = z/3 are skew, and find the distance between these lines.
Find equations of the planes that are parallel to the plane x + 2y - 2z = 1 and two units away from it.
Show that the distance between the parallel planes ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0 is |d1 – d2|| Va? + b2 + c2
Find the distance between the given parallel planes.6z = 4y - 2x, 9z = 1 - 3x + 6y
Find the distance between the given parallel planes.2x - 3y + z = 4, 4x - 6y + 2z = 3
Find the distance from the point to the given plane.(-6, 3, 5), x - 2y - 4z = 8
Find the distance from the point to the given plane.(1, -2, 4), 3x + 2y + 6z = 5
Use the formula in Exercise 12.4.45 to find the distance from the point to the given line.(0, 1, 3); x = 2t, y = 6 - 2t, z = 3 + t
Use the formula in Exercise 12.4.45 to find the distance from the point to the given line.(4, 1, -2); x = 1 + t, y = 3 - 2t, z = 4 - 3t
Which of the following four lines are parallel? Are any of them identical?L1: x = 1 + 6t, y = 1 - 3t, z = 12t + 5L2: x = 1 + 2t, y = t, z = 1 + 4tL3: 2x - 2 = 4 - 4y = z + 1L4: r = (3, 1, 5) + t (4,
Which of the following four planes are parallel? Are any of them identical?P1: 3x + 6y - 3z = 6 P2: 4x - 12y + 8z = 5P3: 9y = 1 + 3x + 6z P4: z − x + 2y - 2
Find parametric equations for the line through the point (0, 1, 2) that is perpendicular to the line x = 1 + t, y = 1 - t, z = 2t and intersects this line.
Find parametric equations for the line through the point (0, 1, 2) that is parallel to the plane x + y + z = 2 and perpendicular to the line x = 1 + t, y = 1 - t, z = 2t.
(a) Find the point at which the given lines intersect:r = (1, 1, 0) + t (1, -1, 2)r = (2, 0, 2) + s(-1, 1, 0)(b) Find an equation of the plane that contains these lines.
Find an equation of the plane with x-intercept a, y-intercept b, and z-intercept c.
Find an equation for the plane consisting of all points that are equidistant from the points (2, 5, 5) and (-6, 3, 1).
Find an equation for the plane consisting of all points that are equidistant from the points (1, 0, -2) and (3, 4, 0).
Find symmetric equations for the line of intersection of the planes.z = 2x - y - 5, z = 4x + 3y - 5
Find symmetric equations for the line of intersection of the planes.5x - 2y - 2z = 1, 4x + y + z = 6
(a) Find parametric equations for the line of intersection of the planes and (b) find the angle between the planes.3x - 2y + z = 1, 2x + y - 3z = 3
(a) Find parametric equations for the line of intersection of the planes and (b) find the angle between the planes.x + y + z = 1, x + 2y + 2z = 1
Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.)5x + 2y + 3z = 2, y = 4x - 6z
Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.)2x - 3y = z, 4x = 3 + 6y + 2z
Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.)x - y + 3z = 1, 3x + y - z = 2
Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.)x + 2y - z = 2, 2x - 2y + z = 1
Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.)9x - 3y + 6z = 2, 2y = 6x + 4z
Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.)x + 4y - 3z = 1, -3x + 6y + 7z = 0
Find the cosine of the angle between the planes x + y + z = 0 and x + 2y + 3z = 1.
Find direction numbers for the line of intersection of the planes x + y + z = 1 and x + z = 0.
Where does the line through (-3, 1, 0) and (-1, 5, 6) intersect the plane 2x + y - z = -2?
Find the point at which the line intersects the given plane.5x = y/2 = z + 2; 10x - 7y + 3z + 24 = 0
Find the point at which the line intersects the given plane.x = t - 1, y = 1 + 2t, z = 3 - t; 3x - y + 2z = 5
Find the point at which the line intersects the given plane.x = 2 - 2t, y = 3t, z = 1 + t; x + 2y - z = 7
Use intercepts to help sketch the plane.6x + 5y - 3z = 15
Use intercepts to help sketch the plane.6x - 3y + 4z = 6
Use intercepts to help sketch the plane.3x + y + 2z = 6
Use intercepts to help sketch the plane.2x + 5y + z = 10
Find an equation of the plane.The plane that passes through the line of intersection of the planes x - z = 1 and y + 2z = 3 and is perpendicular to the plane x + y - 2z = 1
Find an equation of the plane.The plane that passes through the point (1, 5, 1) and is perpendicular to the planes 2x + y - 2z = 2 and x + 3z = 4
Find an equation of the plane.The plane that passes through the points (0, -2, 5) and (-1, 3, 1) and is perpendicular to the plane 2z = 5x + 4y
Find an equation of the plane.The plane that passes through the point (3, 1, 4) and contains the line of intersection of the planes x + 2y + 3z = 1 and 2x - y + z = -3
Find an equation of the plane.The plane that passes through the point (6, -1, 3) andcontains the line with symmetric equations x/3 = y + 4 = z/2
Find an equation of the plane.The plane that passes through the point (3, 5, -1) and contains the line x = 4 - t, y = 2t - 1, z = -3t
Find an equation of the plane.The plane through the points (3, 0, -1), (-2, -2, 3), and (7, 1, -4)
Find an equation of the plane.The plane through the points (2, 1, 2), (3, -8, 6), and (-2, -3, 1)
Find an equation of the plane.The plane through the origin and the points (3, -2, 1) and (1, 1, 1)
Find an equation of the plane.The plane through the points (0, 1, 1), (1, 0, 1), and (1, 1, 0)
Find an equation of the plane.The plane that contains the line x = 1 + t, y = 2 - t, z = 4 - 3t and is parallel to the plane 5x + 2y + z = 1
Find an equation of the plane.The plane through the point (1, 12, 13) and parallel to the plane x + y + z = 0
Find an equation of the plane.The plane through the point (3, -2, 8) and parallel to the plane z = x + y
Find an equation of the plane.The plane through the point (1, -1, -1) and parallel to the plane 5x - y - z = 6
Find an equation of the plane.The plane through the point (2, 0, 1) and perpendicular to the line x = 3t, y = 2 - t, z = 3 + 4t
Find an equation of the plane.The plane through the point (-1, 1/2, 3) and with normal vector i + 4j + k
Find an equation of the plane.The plane through the point (5, 3, 5) and with normal vector 2i + j - k
Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection.L1: x/1 = y - 1/-1 = z - 2/3L2: x - 2/2 = y - 3/-2 = z/7
Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection.L1: x - 2/1 = y - 3/-2 = z - 1/-3L2: x - 3/1 = y + 4/3 = z - 2/-7
Find an equation of the plane.The plane through the origin and perpendicular to the vector (1, -2, 5)
Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection.L1: x = 5 - 12t, y = 3 + 9t, z − + 2 3tL2: x = 3 + 8s, y = -6s, z − 7
Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection.L1: x = 3 + 2t, y = 4 - t, z = 1 + 3tL2: x = 1 + 4s, y = 3 - 2s, z = 4 + 5s
Find parametric equations for the line segment from (-2, 18, 31) to (11, -4, 48).
Find a vector equation for the line segment from (6, -1, 9) to (7, 6, 0).
(a) Find parametric equations for the line through (2, 4, 6) that is perpendicular to the plane x - y + 3z = 7.(b) In what points does this line intersect the coordinate planes?
(a) Find symmetric equations for the line that passes through the point (1, -5, 6) and is parallel to the vector (-1, 2, -3).(b) Find the points in which the required line in part (a) intersects the
Is the line through (-2, 4, 0) and (1, 1, 1) perpendicular to the line through (2, 3, 4) and (3, -1, -8)?
Is the line through (-4, -6, 1) and (-2, 0, -3) parallel to the line through (10, 18, 4) and (5, 3, 14)?
Find parametric equations and symmetric equations for the line.The line of intersection of the planes x + 2y + 3z = 1 and x - y + z = 1
Find parametric equations and symmetric equations for the line.The line through (-6, 2, 3) and parallel to the line 1/2 x = 1/3 y = z + 1
Find parametric equations and symmetric equations for the line.The line through (2, 1, 0) and perpendicular to both i + j and j + k
Find parametric equations and symmetric equations for the line.The line through the points (-8, 1, 4) and (3, -2, 4)
Find parametric equations and symmetric equations for the line.The line through the points (1, 2.4, 4.6) and (2.6, 1.2, 0.3)
Find parametric equations and symmetric equations for the line.The line through the points (0, 1/2, 1) and (2, 1, -3)
Find parametric equations and symmetric equations for the line.The line through the origin and the point (4, 3, -1)
Find a vector equation and parametric equations for the line.The line through the point (1, 0, 6) and perpendicular to the plane x + 3y + z = 5
Find a vector equation and parametric equations for the line.The line through the point (0, 14, -10) and parallel to the line x = -1 + 2t, y = 6 - 3t, z = 3 + 9t
Find a vector equation and parametric equations for the line.The line through the point (2, 2.4, 3.5) and parallel to the vector 3i + 2j - k
Find a vector equation and parametric equations for the line.The line through the point (6, -5, 2) and parallel to the vector (1, 3, -2/3)
If v1, v2, and v3 are non-coplanar vectors, let(These vectors occur in the study of crystallography. Vectors of the form n1v1 + n2v2 + n3v3, where each ni is an integer, form a lattice for a crystal.
Suppose that a ≠ 0.(a) If a • b = a • c, does it follow that b = c?(b) If a x b = a x c, does it follow that b = c?(c) If a • b = a • c and a x b = a x c, does it follow that b = c?
Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS.P(3, 0, 1), Q(-1, 2, 5), R(5, 1, -1), S(0, 4, 2)
Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS.P(-2, 1, 0), Q(2, 3, 2), R(1, 4, -1), S(3, 6, 1)
Find the volume of the parallelepiped determined by the vectors a, b, and c.a = i + j , b = j + k, c = i + j + k
Find the volume of the parallelepiped determined by the vectors a, b, and c.a = (1, 2, 3), b = (-1, 1, 2), c = (2, 1, 4)
(a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and (b) find the area of triangle PQR.(b) P(2, -3, 4), Q(-1, -2, 2), R(3, 1, -3)
(a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and (b) find the area of triangle PQR.(b) P(0, -2, 0), Q(4, 1, -2), R(5, 3, 1)
(a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and (b) find the area of triangle PQR.(b) P(0, 0, -3), Q(4, 2, 0), R(3, 3, 1)
(a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and (b) find the area of triangle PQR.(b) P(1, 0, 1), Q(-2, 1, 3), R(4, 2, 5)
Find the area of the parallelogram with vertices P(1, 0, 2), Q(3, 3, 3), R(7, 5, 8), and S(5, 2, 7).
Find the area of the parallelogram with vertices A(-3, 0), B(-1, 3), C(5, 2), and D(3, -1).
Prove the property of cross products (Theorem 11).Property 4: (a + b) x c = a x c + b x c
Prove the property of cross products (Theorem 11).Property 3: a x (b + c) = a x b + a x c
Prove the property of cross products (Theorem 11).Property 2: (ca) x b = c(a x b) = a x (cb)

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