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mathematics
precalculus
Thomas Calculus Early Transcendentals 13th Edition Joel R Hass, Christopher E Heil, Maurice D Weir - Solutions
Sketch the surfaces.9x2 + 4y2 + 36z2 = 36
Find parametrizations for the line segments joining the points. Draw coordinate axes and sketch each segment, indicating the direction of increasing t for your parametrization.(1, 0, -1), (0, 3, 0)
Prove that a parallelogram is a rectangle if and only if its diagonals are equal in length.
Verify that (u * v) · w = (v * w) · u = (w * u) · v and find the volume of the parallelepiped (box) determined by u, v, and w. u 2i + j V 2i - j + k W i + 2k
Use vectors to prove thatfor any four numbers a, b, c, and d. (a² + b²) (c² + d²) = (ac + bd)²
Express each vector in the form v = v1i + v2j + v3k.-2u + 3v if u = (-1, 0, 2) and v = (1, 1, 1)
Show that the indicated diagonal of the parallelogram determined by vectors u and v bisects the angle between u and v if |u| = |v|. u V
Verify that (u * v) · w = (v * w) · u = (w * u) · v and find the volume of the parallelepiped (box) determined by u, v, and w. u i + j - 2k V -i-k W 2i + 4j - 2k
Copy vectors u, v, and w head to tail as needed to sketch the indicated vector.a. u +v b. u + v + wc. u -v d. u - w u W V
Draw coordinate axes and then sketch u, v, and u * v as vectors at the origin.u = i, v = i + j
Describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.a. 1 ≤ x2 + y2 + z2 ≤ 4b. x2 + y2 + z2 ≤ 1, z ≥ 0
Sketch the surfaces.z = x2 + 4y2
Find equations for the planes.The plane through P0(0, 2, -1) normal to n = 3i - 2j - k
Show that dot multiplication of vectors is positive definite; that is, show u · u ≥ 0 for every vector u and that u · u = 0 if and only if u = 0.
Suppose that a box is being towed up an inclined plane as shown in the figure. Find the force w needed to make the component of the force parallel to the inclined plane equal to 2.5 lb. 15° 33° W
Draw coordinate axes and then sketch u, v, and u * v as vectors at the origin.u = i - j, v = i + j
Describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.a. x = y, z = 0 b. x = y, no restriction on z
Copy vectors u, v, and w head to tail as needed to sketch the indicated vector.a. u -v b. u - v + wc. 2u - v d. u + v + w W n
Sketch the surfaces.z = 8 - x2 - y2
Find equations for the planes.The plane through (1, -1, 3) parallel to the plane 3x + y + z = 7
A gun with muzzle velocity of 1200 ft / sec is fired at an angle of 8° above the horizontal. Find the horizontal and vertical components of the velocity.
Show that |u + v| ≤ |u| + |v| for any vectors u and v.
If |v| = 2, |w| = 3, and the angle between v and w is π/3, find |v - 2w|.
Find the magnitude of the torque exerted by F on the bolt at P ifand |F| = 30 lb . Answer in footpounds. |PQ| = 8 in.
Let u = 5i - j + k, v = j - 5k, w = -15i + 3j - 3k. Which vectors, if any, are (a) Perpendicular? (b) Parallel? Give reasons for your answers.
Describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.a. y ≥ x2, z ≥ 0 b. x ≤ y2, 0 ≤ z ≤ 2
Sketch the surfaces.x = 4 - 4y2 - z2
Find equations for the planes.The plane through (1, 1, -1), (2, 0, 2), and (0, -2, 1)
Show that w = |v| u + |u| v bisects the angle between u and v.
Find the magnitude of the torque exerted by F on the bolt at P ifand |F| = 30 lb . Answer in footpounds. |PQ| = 8 in.
For what value or values of a will the vectors u = 2i + 4j - 5k and v = -4i - 8j + ak be parallel?
Let u = i + 2j - k, v = -i + j + k, w = i + k, r = -(π/2)i - πj + (π/2)k. Which vectors, if any, are (a) Perpendicular? (b) Parallel? Give reasons for your answers.
Describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.a. z = 1 - y, no restriction on xb. z = y3, x = 2
Find equations for the planes.The plane through (2, 4, 5), (1, 5, 7), and (-1, 6, 8)
Sketch the surfaces.y = 1 - x2 - z2
Express each vector as a product of its length and direction.2i + j - 2k
a. Since u · v = |u ||v| cos θ, show that the inequality |u · v| ≤ |u||v| holds for any vectors u and v.b. Under what circumstances, if any, does |u · v| equal |u||v|? Give reasons for your answer.
Show that |v| u + |u| v and |v| u - |u| v are orthogonal.
Express each vector as a product of its length and direction. 3 + 45
Describe the given set with a single equation or with a pair of equations.The plane perpendicular to thea. x-axis at (3, 0, 0) b. y-axis at (0, -1, 0)c. z-axis at (0, 0, -2)
Sketch the surfaces.x2 + y2 = z2
Find equations for the planes.The plane through P0(2, 4, 5) perpendicular to the line x = 5 + t, y = 1 + 3t, z = 4t
Express each vector as a product of its length and direction.9i - 2j + 6k
Show that dot multiplication of vectors is positive definite; that is, show that u · u ≥ 0 for every vector u and that u · u = 0 if and only if u = 0.
Describe the given set with a single equation or with a pair of equations.The plane through the point (3, -1, 2) perpendicular to thea. x-axis b. y-axis c. z-axis
Express each vector as a product of its length and direction. √6 i 1 V 1 6 k
Sketch the surfaces.4x2 + 9z2 = 9y2
Find equations for the planes.The plane through A(1, -2, 1) perpendicular to the vector from the origin to A
Express each vector as a product of its length and direction.5k
If u1 and u2 are orthogonal unit vectors and v = au1 + bu2, find v · u1 .
Describe the given set with a single equation or with a pair of equations.The plane through the point (3, -1, 1) parallel to thea. xy-plane b. yz-plane c. xz-plane
Sketch the surfaces.x2 + y2 - z2 = 1
Find the point of intersection of the lines x = 2t + 1, y = 3t + 2, z = 4t + 3, and x = s + 2, y = 2s + 4, z = -4s - 1, and then find the plane determined by these lines.
In real-number multiplication, if uv1 = uv2 and u ≠ 0, we can cancel the u and conclude that v1 = v2 . Does the same rule hold for the dot product? That is, if u · v1 = u · v2 and u ≠ 0, can you conclude that v1 = v2? Give reasons for your answer.
Find the plane containing the intersecting lines. Ll: x = -1 + t, L2: x = 1 4s, y = 2 + t, z = 1- t; -∞ < t
Express each vector as a product of its length and direction. i √3 + j √3 + k √3
Describe the given set with a single equation or with a pair of equations.The circle of radius 2 centered at (0, 0, 0) and lying in thea. xy-plane b. yz-plane c. xz-plane
Find the point of intersection of the lines x = t, y = -t + 2, z = t + 1, and x = 2s + 2, y = s + 3, z = 5s + 6, and then find the plane determined by these lines.
Using the definition of the projection of u onto v, show by direct calculation that (u - projv u) · projv u = 0.
Describe the given set with a single equation or with a pair of equations.The circle of radius 2 centered at (0, 2, 0) and lying in thea. xy-plane b. yz-plane c. plane y = 2
A force F = 2i + j - 3k is applied to a spacecraft with velocity vector v = 3i - j . Express F as a sum of a vector parallel to v and a vector orthogonal to v.
Find the plane containing the intersecting lines. Ll: x = 1, L2: x = 1 +s, y = y = 3 - 33t, 3t, z = z = -2- t; -2- t; -∞ < t
Find the vectors whose lengths and directions are given. Try to do the calculations without writing. Length a. 7 b. √2 C. 13 12 d. a > 0 Direction −j 3 i 5 12 31-11-1k i k i+ /رایت - 1 97/2 √6 k
Show that v = ai + bj is perpendicular to the line ax + by = c by establishing that the slope of the vector v is the negative reciprocal of the slope of the given line.
Find the vectors whose lengths and directions are given. Try to do the calculations without writing. Length a. 2 b. √3 1 2 d. 7 C. Direction i -k 3 + k ²7/₁ + k 7
Show that the vector v = ai + bj is parallel to the line bx - ay = c by establishing that the slope of the line segment representing v is the same as the slope of the given line.
Find a plane through P0(2, 1, -1) and perpendicular to the line of intersection of the planes 2x + y - z = 3, x + 2y + z = 2.
Sketch the surfaces.y2 - x2 = z
Describe the given set with a single equation or with a pair of equations.The line through the point (1, 3, -1) parallel to thea. x-axis b. y-axis c. z-axis
Let u, v, and w be vectors. Which of the following make sense, and which do not? Give reasons for your answers.a. (u * v) · wb. u * (v · w)c. u * (v * w)d. u · (v · w)
Describe the given set with a single equation or with a pair of equations.The circle of radius 1 centered at (-3, 4, 1) and lying in a plane parallel to thea. xy-plane b. yz-plane c. xz-plane
Sketch the surfaces.(y2/4) - (x2/4) - z2 = 1
Show that except in degenerate cases, (u * v) * w lies in the plane of u and v, whereas u * (v * w) lies in the plane of v and w. What are the degenerate cases?
Describe the given set with a single equation or with a pair of equations.The set of points in space equidistant from the origin and the point (0, 2, 0)
Sketch the surfaces.x2 - y2 = z
Find a plane through the points P1(1, 2, 3), P2(3, 2, 1) and perpendicular to the plane 4x - y + 2z = 7.
Use the result of Exercise 31 to find an equation for the line through P perpendicular to v. Then sketch the line. Include v in your sketch as a vector starting at the origin.P(2, 1), v = i + 2jExercise 31Show that v = ai + bj is perpendicular to the line ax + by = c by establishing that the slope
Find a vector of magnitude 7 in the direction of v = 12i - 5k.
If u * v = u * w and u ≠ 0, then does v = w? Give reasons for your answer.
Describe the given set with a single equation or with a pair of equations.The circle in which the plane through the point (1, 1, 3) perpendicular to the z-axis meets the sphere of radius 5 centered at the origin
Finda. The direction of andb. The midpoint of line segment P1P2.P1(-1, 1, 5) P2(2, 5, 0) P₁P₂
Sketch the surfaces.z = 1 + y2 - x2
Find the distance from the point to the line.(0, 0, 12); x = 4t, y = -2t, z = 2t
Use the result of Exercise 31 to find an equation for the line through P perpendicular to v. Then sketch the line. Include v in your sketch as a vector starting at the origin.P(-1, 2), v = -2i - jExercise 31Show that v = ai + bj is perpendicular to the line ax + by = c by establishing that the
Finda. The direction of andb. The midpoint of line segment P1P2.P1(1, 4, 5) P2(4, -2, 7) P₁P₂
Find a vector of magnitude 3 in the direction opposite to the direction of v = (1/2)i - (1/2)j - (1/2)k.
If u ≠ 0 and if u * v = u * w and u # v = u # w, then does v = w? Give reasons for your answer.
Describe the given set with a single equation or with a pair of equations.The set of points in space that lie 2 units from the point (0, 0, 1) and, at the same time, 2 units from the point (0, 0, -1)
Sketch the surfaces.4x2 + 4y2 = z2
Find the distance from the point to the line.(0, 0, 0); x = 5 + 3t, y = 5 + 4t, z = -3 - 5t
Use the result of Exercise 31 to find an equation for the line through P perpendicular to v. Then sketch the line. Include v in your sketch as a vector starting at the origin.P(-2, -7), v = -2i + jExercise 31Show that v = ai + bj is perpendicular to the line ax + by = c by establishing that the
Finda. The direction of andb. The midpoint of line segment P1P2.P1(3, 4, 5) P2(2, 3, 4) P₁P₂
Find the areas of the parallelograms whose vertices are given.A(1, 0), B(0, 1), C(-1, 0), D(0, -1)
Write inequalities to describe the set.The slab bounded by the planes z = 0 and z = 1
Sketch the surfaces.y = -(x2 + z2)
Find the distance from the point to the line.(2, 1, 3); x = 2 + 2t, y = 1 + 6t, z = 3
Use the result of Exercise 31 to find an equation for the line through P perpendicular to v. Then sketch the line. Include v in your sketch as a vector starting at the origin.P(11, 10), v = 2i - 3jExercise 31Show that v = ai + bj is perpendicular to the line ax + by = c by establishing that the
Find the areas of the parallelograms whose vertices are given.A(0, 0), B(7, 3), C(9, 8), D(2, 5)
Write inequalities to describe the set.The solid cube in the first octant bounded by the coordinate planes and the planes x = 2, y = 2, and z = 2
Finda. The direction of andb. The midpoint of line segment P1P2.P1(0, 0, 0) P2(2, -2, -2) P₁P₂
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