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mathematics
calculus 6th edition
Calculus 6th Edition James Stewart - Solutions
Find the cross product a × b and verify that it is orthogonal to both a and b.〈6, 0, –2〉, b = 〈0, 8, 0〉
Find the cross product a × b and verify that it is orthogonal to both a and b.a = 〈1, 1, –1〉, b = 〈2, 4, 6〉
Sketch the points (0, 5, 2), (4, 0, –1), (2, 4, 6), and (1, –1, 2) on a single set of coordinate axes.
Find the dot product of two vectors if their lengths are 6 and 1/3 and the angle between them is π/4.
Find the cross product a × b and verify that it is orthogonal to both a and b.a = i + 3j – 2k, b = –i + 5k
Find a · b. a = (-2,3), b= (-5,12)
Which of the points P(6, 2, 3), Q(–5, –1, 4), and R(0, 3, 8) is closest to the xz-plane? Which point lies in the yz-plane?
Describe and sketch the surface.y2 + 4z2 = 4
Find the cross product a × b and verify that it is orthogonal to both a and b.a = j + 7k, b = 2i – j + 4k
Find a · b.a = 〈–2,3〉, b = 〈0.7, 1.2〉
Describe and sketch the surface.z = 4 – x2
Find a · b. a =(4, 1₁), b = (6,-3, -8)
Find the cross product a × b and verify that it is orthogonal to both a and b.a = i – j – k, b = 1/2i + j + 1/2k
Describe and sketch the surface.x – y2 = 0
Find parametric equations and symmetric equations for the line.The line through the origin and the point (1, 2, 3)
Find the cross product a × b and verify that it is orthogonal to both a and b.la = i + etj + e–tk, b = 2i + etj – e–tk
Find a · b.a = 〈s, 2s, 3s), b = 〈t. –t. 5t〉
Describe and sketch the surface.yz = 4
Find a vector a with representation given by the directed line segment AB(vector). Draw AB(vector) and the equivalent representation starting at the origin.A(2, 3), B(–2, 1)
Find parametric equations and symmetric equations for the line.The line through the points (1,3, 2) and (–4, 3, 0)
Find a · b.a = i – 2j + 3k, b = 5i + 9k
Describe and sketch the surface.z = cos x
Find a vector a with representation given by the directed line segment AB(vector). Draw AB(vector) and the equivalent representation starting at the origin.A(–2, –2), B(5, 3)
Find parametric equations and symmetric equations for the line.The line through the points (6, 1, –3) and (2, 4, 5)
Find a · b.a = 4j – 3k, b = 2i + 4j + 6k
Describe and sketch the surface.x2 – y2 = 1
Find a · b. |a|= 6, |b|= 5, the angle between a and b is 27/3
Find a vector a with representation given by the directed line segment AB(vector). Draw AB(vector) and the equivalent representation starting at the origin.A(–1, 3), B(2, 2)
Find a · b. |a|=3, |b|= √6, the angle between a and b is 45°
Find a vector a with representation given by the directed line segment AB(vector). Draw AB(vector) and the equivalent representation starting at the origin.A(2, 1), B(0, 6)
Find parametric equations and symmetric equations for the line.The line through (1, –1, 1) and parallel to the line x + 2 = 1/2y = z – 3
Find an equation of the sphere with center (1, –4, 3) and radius 5. What is the intersection of this sphere with the xz-plane?
Find a vector a with representation given by the directed line segment AB(vector). Draw AB(vector) and the equivalent representation starting at the origin.A(4, 0, –2), B(4, 2, 1)
Find parametric equations and symmetric equations for the line.The line of intersection of the planes x + y + z = 1 and x + z = 0
Use traces to sketch and identify the surface.9x2 – y2 + z2 = 0
Use traces to sketch and identify the surface.x2 = y2 + 4z2
Find the sum of the given vectors and illustrate geometrically.〈–2, –1〉, 〈5, 7〉
Is the line through (4, 1, –1) and (2, 5, 3) perpendicular to the line through (–3, 2, 0) and (5, 1, 4)?
A street vendor sells a hamburgers, b hot dogs, and c soft drinks on a given day. He charges $2 for a hamburger, $1.50 for a hot dog, and $1 for a soft drink. If A = 〈a, b, c〉 and P = 〈2, 1.5, 1〉, what is the meaning of the dot product A · P?
Use traces to sketch and identify the surface.25x2 + 4y2 + z2 = 100
Find the sum of the given vectors and illustrate geometrically.〈0, 1, 2〉, 〈0, 0, –3〉
Show that the equation represents a sphere, and find its center and radius.x2 + y2 + z2 – 6x + 4y – 2z = 11
Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.)ka = 〈–8, 6〉, b = 〈√7, 3〉
Use traces to sketch and identify the surface.–x2 + 4y2 – z2 = 4
Find the sum of the given vectors and illustrate geometrically.〈–1, 0, 2〉, 〈0, 4, 0〉
Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.)a = 〈√3, 1), b = 〈0, 5〉
Use traces to sketch and identify the surface.4x2 + 9y2 + z = 0
Use traces to sketch and identify the surface.36x2 + y2 + 36z2 = 36
Find a vector equation for the line segment from (2, –1, 4) to (4, 6, 1).
Find a + b, 2a + 3b, |a|, and |a – b|.a = 〈5, –12〉, b = 〈–3, –6〉
If a = 〈1, 2, 1〉 and b = 〈0, 1,3〉, find a × b and b × a.
Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.)a = 〈3, –1, 5〉, b = 〈–2, 4, 3〉
Use traces to sketch and identify the surface.4x2 – 16y2 + z2 = 16
Find parametric equations for the line segment from (10, 3, 1) to (5, 6, –3).
Find a + b, 2a + 3b, |a|, and |a – b|.a = 4i + j, b = i – 2j
If a = 〈3, 1, 2〉, b = 〈–1, 1, 0〉, and c = 〈0, 0, –4〉 show that a × (b × c) = (a × b) ≠ (a × b) × c.
Show that the equation represents a sphere, and find its center and radius.4x2 + 4y2 + 4z2 – 8x + 16y = 1
Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.)a = 〈4, 0, 2〉. b = 〈2, –1, 0〉
Find a + b, 2a + 3b, |a|, and |a – b|.a = i + 2j – 3k, b = –2i – j + 5k
Find two unit vectors orthogonal to both 〈1, –1, 1〉 and 〈0, 4, 4〉.
Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.)a = j + k, b = i + 2j – 3k
Find a + b, 2a + 3b, |a|, and |a – b|.a = 2i – 4j + 4k, b = 2j – k
Find two unit vectors orthogonal to both i + j + k and 2i + k.
Determine whether the given vectors are orthogonal, parallel, or neither. (a) a = (-5,3,7), b (6,-8,2) = (b) a = (4,6), b= (-3,2) (c) a = -i +2j+5k, b=3i+4j - k (d) a = 2i + 6j - 4k, b = -3i - 9j + 6k
Find an equation of a sphere if one of its diameters has end- points (2, 1, 4) and (4, 3, 10).
Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.)ka = i + 2j – 2k, b = 4i – 3k
Find a unit vector that has the same direction as the given vector.–3i + 7j
Find, correct to the nearest degree, the three angles of the triangle with the given vertices.A(1,0), B(3, 6), C(–1, 4)
Find, correct to the nearest degree, the three angles of the triangle with the given vertices.D(0, 1, 1), E(–2, 4, 3). F(1, 2, –1)
Find a unit vector that has the same direction as the given vector.〈–4, 2, 4〉
Find an equation of the plane.The plane through the point (6, 3, 2) and perpendicular to the vector (–2, 1, 5)
Determine whether the given vectors are orthogonal, parallel, or neither. (a) u = (-3,9,6), v= (4, -12, -8) (b) u = i- j + 2k, v=2i-j + k (c) u = (a,b,c), v = (-b₁a,0)
Describe in words the region of R3 represented by the equation or inequality.ky = –4
Find an equation of the plane.The plane through the point (4, 0, –3) and with normal vector j + 2k
Find a vector that has the same direction as 〈–2, 4, 2〉 but has length 6.
Describe in words the region of R3 represented by the equation or inequality.x = 10
Find an equation of the plane.The plane through the point (1, –1, 1) and with normal vector i + j – k
Describe in words the region of R3 represented by the equation or inequality.x > 3
For what values of b are the vectors 〈–6, b, 2〉 and 〈b, b2, b〉 orthogonal?
Find an equation of the plane.The plane through the point (–2, 8, 10) and perpendicular to the line x = 1 + t, y = 2t, z = 4 – 3t
Describe in words the region of R3 represented by the equation or inequality.y ≥ 0
Find the area of the parallelogram with vertices A(–2, 1). B(0, 4), C(4, 2), and D(2, –1).
Describe in words the region of R3 represented by the equation or inequality.0 ≤ z ≤ 6
Find an equation of the plane.The plane through the point (–1, 6, –5) and parallel to the plane x + y + z +2 = 0
Find the area of the parallelogram with vertices K(1, 2, 3). L(1, 3, 6), M(3, 8, 6), and N(3, 7, 3).
Describe in words the region of R3 represented by the equation or inequality.z2 = 1
Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.)〈3, 4, 5〉
Reduce the equation to one of the standard forms, classify the surface, and sketch it.z2 = 4x2 + 9y2 + 36
Find an equation of the plane.The plane through the point (4, –2, 3) and parallel to the plane 3x – 7z = 12
(a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R, (b) Find the area of triangle PQR.P(1, 0, 0), Q(0, 2, 0), R(0, 0, 3)
Describe in words the region of R3 represented by the equation or inequality.x2 + y2 + z2 ≤ 3
Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.)〈1, –2, –1〉
Reduce the equation to one of the standard forms, classify the surface, and sketch it.x2 = 2y2 + 3z2
Find an equation of the plane.The plane that contains the line x = 3 + 2t, y = t, z = 8 – t and is parallel to the plane 2x + 4y + 8z = 17
(a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and (b) Find the area of triangle PQR.P(2, 1, 5), Q(–1, 3, 4), R(3, 0, 6)
Describe in words the region of R3 represented by the equation or inequality.x = z
Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.)2i + 3j – 6k
Identify and sketch the graph of each surface.x2 = y2 + 4z2
Reduce the equation to one of the standard forms, classify the surface, and sketch it.x = 2y2 + 3z2
Describe in words the region of R3 represented by the equation or inequality.x2 + z2 ≤ 9
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