Calculus (MindTap Course List)
8th Edition
ISBN: 9781285740621
Author: James Stewart
Publisher: Cengage Learning
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Chapter 12.4, Problem 1E
To determine
To find:
(i) The cross product
(ii) Verify if the cross product is orthogonal to
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Exercise 11.3 A slope field is given for the equation y' = 4y+4.
(a) Sketch the particular solution that corresponds to y(0) = −2
(b) Find the constant solution
(c) For what initial conditions y(0) is the solution increasing?
(d) For what initial conditions y(0) is the solution decreasing?
(e) Verify these results using only the differential equation y' = 4y+4.
Chapter 12 Solutions
Calculus (MindTap Course List)
Ch. 12.1 - Suppose you start at the origin, move along the...Ch. 12.1 - Sketch the points (1,5,3),(0,2,3),(3,0,2), and...Ch. 12.1 - Which of the points A(4,0,1),B(3,1,5), and...Ch. 12.1 - What are the projections of the point 2, 3, 5 on...Ch. 12.1 - What does the equation x=4 represent in 2? What...Ch. 12.1 - What does the equation y = 3 represent in 3? What...Ch. 12.1 - Describe and sketch the surface in 3 represented...Ch. 12.1 - Prob. 8ECh. 12.1 - Find the lengths of the sides of the triangle PQR....Ch. 12.1 - Find the lengths of the sides of the triangle PQR....
Ch. 12.1 - Determine whether the points lie on a straight...Ch. 12.1 - Prob. 12ECh. 12.1 - Find an equation of the sphere with centre (3,2,5)...Ch. 12.1 - Find an equation of the sphere with centre (2,6,4)...Ch. 12.1 - Find an equation of the sphere that passes through...Ch. 12.1 - Prob. 16ECh. 12.1 - Prob. 17ECh. 12.1 - Prob. 18ECh. 12.1 - Prob. 19ECh. 12.1 - Prob. 20ECh. 12.1 - a Prove that the midpoint of the line segment from...Ch. 12.1 - Find an equation of a sphere if one of its...Ch. 12.1 - Prob. 23ECh. 12.1 - Find an equation of the largest sphere with centre...Ch. 12.1 - Prob. 25ECh. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Prob. 27ECh. 12.1 - Prob. 28ECh. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Prob. 30ECh. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Prob. 33ECh. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Prob. 36ECh. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Prob. 38ECh. 12.1 - Prob. 39ECh. 12.1 - Write inequalities to describe the region. The...Ch. 12.1 - Write inequalities to describe the region. The...Ch. 12.1 - Write inequalities to describe the region. The...Ch. 12.1 - The figure shows a line L1 in space and a second...Ch. 12.1 - Consider the points P such that the distance from...Ch. 12.1 - Prob. 45ECh. 12.1 - Find the volume of the solid that lies inside both...Ch. 12.1 - Find the distance between the spheres x2+y2+z2=4...Ch. 12.1 - Describe and sketch a solid with the following...Ch. 12.2 - Are the following quantities vectors or scalars?...Ch. 12.2 - What is the relationship between the point 4, 7...Ch. 12.2 - Name all the equal vectors in the parallelogram...Ch. 12.2 - Prob. 4ECh. 12.2 - Prob. 5ECh. 12.2 - Prob. 6ECh. 12.2 - In the figure, the tip of c and the tail of d are...Ch. 12.2 - If the vectors in the figure satisfy |u|=|v|=1 and...Ch. 12.2 - Find a vector a with representation given by the...Ch. 12.2 - Prob. 10ECh. 12.2 - Prob. 11ECh. 12.2 - Prob. 12ECh. 12.2 - Prob. 13ECh. 12.2 - Prob. 14ECh. 12.2 - Prob. 15ECh. 12.2 - Prob. 16ECh. 12.2 - Prob. 17ECh. 12.2 - Prob. 18ECh. 12.2 - Prob. 19ECh. 12.2 - Find a+b,4a+2b,|a|, and |ab|. a=5i+3j,b=i2jCh. 12.2 - Prob. 21ECh. 12.2 - Find a+b,4a+2b,|a|, and |ab|. a=8,1,4,b=5,2,1Ch. 12.2 - Prob. 23ECh. 12.2 - Prob. 24ECh. 12.2 - Prob. 25ECh. 12.2 - Prob. 26ECh. 12.2 - What is the angle between the given vector and the...Ch. 12.2 - Prob. 28ECh. 12.2 - If v lies in the first quadrant and makes an angle...Ch. 12.2 - Prob. 30ECh. 12.2 - Prob. 31ECh. 12.2 - Find the magnitude of the resultant force and the...Ch. 12.2 - Find the magnitude of the resultant force and the...Ch. 12.2 - Prob. 34ECh. 12.2 - Prob. 35ECh. 12.2 - Prob. 36ECh. 12.2 - A block-and-tackle pulley hoist is suspended in a...Ch. 12.2 - The tension T at each end of a chain has magnitude...Ch. 12.2 - A boatman wants to cross a canal that is 3 km wide...Ch. 12.2 - Three forces act on an object. Two of the forces...Ch. 12.2 - Prob. 41ECh. 12.2 - Prob. 42ECh. 12.2 - If A, B, and C are the vertices of a triangle,...Ch. 12.2 - Prob. 44ECh. 12.2 - a Draw the vectors a=3,2,b=2,1 and c=7,1. b Show,...Ch. 12.2 - Prob. 46ECh. 12.2 - Prob. 47ECh. 12.2 - Prob. 48ECh. 12.2 - Prob. 49ECh. 12.2 - Prove Property 5 of vectors algebraically for the...Ch. 12.2 - Use vectors to prove that the line joining the...Ch. 12.2 - Prob. 52ECh. 12.3 - Which of the following expressions are meaningful?...Ch. 12.3 - Prob. 2ECh. 12.3 - 210 Find ab. a=1.5,0.4, b=4,6Ch. 12.3 - Prob. 4ECh. 12.3 - Prob. 5ECh. 12.3 - Prob. 6ECh. 12.3 - Prob. 7ECh. 12.3 - Prob. 8ECh. 12.3 - Prob. 9ECh. 12.3 - Prob. 10ECh. 12.3 - If u is a unit vector, find uv and uw.Ch. 12.3 - If u is a unit vector, find uv and uw.Ch. 12.3 - Prob. 13ECh. 12.3 - Prob. 14ECh. 12.3 - Prob. 15ECh. 12.3 - Find the angle between the vectors. First find an...Ch. 12.3 - Prob. 17ECh. 12.3 - Prob. 18ECh. 12.3 - Prob. 19ECh. 12.3 - Prob. 20ECh. 12.3 - 2122 Find, correct to the nearest degree, the...Ch. 12.3 - Prob. 22ECh. 12.3 - Prob. 23ECh. 12.3 - 2324 Determine whether the given vectors are...Ch. 12.3 - Prob. 25ECh. 12.3 - Prob. 26ECh. 12.3 - Prob. 27ECh. 12.3 - Prob. 28ECh. 12.3 - Prob. 29ECh. 12.3 - Prob. 30ECh. 12.3 - 3132 Find the acute angles between the curves at...Ch. 12.3 - 3132 Find the acute angles between the curves at...Ch. 12.3 - Prob. 33ECh. 12.3 - Prob. 34ECh. 12.3 - Prob. 35ECh. 12.3 - Prob. 36ECh. 12.3 - Prob. 37ECh. 12.3 - Prob. 38ECh. 12.3 - 3944 Find the scalar and vector projections of b...Ch. 12.3 - Prob. 40ECh. 12.3 - Prob. 41ECh. 12.3 - Find the scalar and vector projections of b onto...Ch. 12.3 - Find the scalar and vector projections of b onto...Ch. 12.3 - Prob. 44ECh. 12.3 - Prob. 45ECh. 12.3 - Prob. 46ECh. 12.3 - If a=3,0,1, find a vector b such that compab=2.Ch. 12.3 - Prob. 48ECh. 12.3 - Prob. 49ECh. 12.3 - Prob. 50ECh. 12.3 - Prob. 51ECh. 12.3 - Prob. 52ECh. 12.3 - Use a scalar projection to show that the distance...Ch. 12.3 - If r=x,y,z,a=a1,a2,a3, and b=b1,b2,b3, show that...Ch. 12.3 - Prob. 55ECh. 12.3 - Prob. 56ECh. 12.3 - Prob. 57ECh. 12.3 - If c=|a|b+|b|a, where a, b, and c are all nonzero...Ch. 12.3 - Prob. 59ECh. 12.3 - Prob. 60ECh. 12.3 - Use Theorem 3 to prove the Cauchy-Schwarz...Ch. 12.3 - Prob. 62ECh. 12.3 - The Parallelogram Law states that...Ch. 12.3 - Prob. 64ECh. 12.3 - Prob. 65ECh. 12.4 - Prob. 1ECh. 12.4 - Prob. 2ECh. 12.4 - 17 Find the cross product ab and verify that it is...Ch. 12.4 - 17 Find the cross product ab and verify that it is...Ch. 12.4 - Prob. 5ECh. 12.4 - Prob. 6ECh. 12.4 - Prob. 7ECh. 12.4 - If a=i2k and b=j+k, find ab. Sketch a, b, and ab....Ch. 12.4 - Prob. 9ECh. 12.4 - 912 Find the vector, not with determinants, but by...Ch. 12.4 - 912 Find the vector, not with determinants, but by...Ch. 12.4 - 912 Find the vector, not with determinants, but by...Ch. 12.4 - Prob. 13ECh. 12.4 - Prob. 14ECh. 12.4 - Prob. 15ECh. 12.4 - Prob. 16ECh. 12.4 - Prob. 17ECh. 12.4 - If a=1,0,1,b=2,1,1, and c=0,1,3, show that...Ch. 12.4 - Find two unit vectors orthogonal to both 3,2,1 and...Ch. 12.4 - Prob. 20ECh. 12.4 - Prob. 21ECh. 12.4 - Prob. 22ECh. 12.4 - Prob. 23ECh. 12.4 - Prob. 24ECh. 12.4 - Prob. 25ECh. 12.4 - Prob. 26ECh. 12.4 - Prob. 27ECh. 12.4 - Find the area of the parallelogram with vertices...Ch. 12.4 - 2932 a Find a nonzero vector orthogonal to the...Ch. 12.4 - 2932 a Find a nonzero vector orthogonal to the...Ch. 12.4 - 2932 a Find a nonzero vector orthogonal to the...Ch. 12.4 - Prob. 32ECh. 12.4 - Prob. 33ECh. 12.4 - Prob. 34ECh. 12.4 - Prob. 35ECh. 12.4 - Prob. 36ECh. 12.4 - Prob. 37ECh. 12.4 - Prob. 38ECh. 12.4 - Prob. 39ECh. 12.4 - Prob. 40ECh. 12.4 - A wrench 30 cm long lies along the positive y-axis...Ch. 12.4 - Prob. 42ECh. 12.4 - Prob. 43ECh. 12.4 - a Find all vectors v such that 1,2,1v=3,1,5 b...Ch. 12.4 - Prob. 45ECh. 12.4 - a Let P be a point not on the plane that passes...Ch. 12.4 - Prob. 47ECh. 12.4 - Prob. 48ECh. 12.4 - Prob. 49ECh. 12.4 - Prob. 50ECh. 12.4 - Prob. 51ECh. 12.4 - Prob. 52ECh. 12.4 - Suppose that a0. a If ab=ac, does it follow that...Ch. 12.4 - If v1, v2, and v3 are noncoplanar vectors, let...Ch. 12.5 - Determine whether each statement is true or false...Ch. 12.5 - Prob. 2ECh. 12.5 - Prob. 3ECh. 12.5 - Prob. 4ECh. 12.5 - Prob. 5ECh. 12.5 - 612 Find parametric equations and symmetric...Ch. 12.5 - Prob. 7ECh. 12.5 - Prob. 8ECh. 12.5 - Prob. 9ECh. 12.5 - 612 Find parametric equations and symmetric...Ch. 12.5 - 612 Find parametric equations and symmetric...Ch. 12.5 - 612 Find parametric equations and symmetric...Ch. 12.5 - Prob. 13ECh. 12.5 - Is the line through (2,4,0) and (1,1,1)...Ch. 12.5 - a Find symmetric equations for the line that...Ch. 12.5 - a Find parametric equations for the line through...Ch. 12.5 - Prob. 17ECh. 12.5 - Prob. 18ECh. 12.5 - 1922 Determine whether the lines L1 and L2 are...Ch. 12.5 - Prob. 20ECh. 12.5 - Prob. 21ECh. 12.5 - Prob. 22ECh. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - Prob. 40ECh. 12.5 - Prob. 41ECh. 12.5 - Prob. 42ECh. 12.5 - Prob. 43ECh. 12.5 - Prob. 44ECh. 12.5 - 4547 Find the point at which the line intersects...Ch. 12.5 - Prob. 46ECh. 12.5 - Prob. 47ECh. 12.5 - Prob. 48ECh. 12.5 - Find direction numbers for the line of...Ch. 12.5 - Prob. 50ECh. 12.5 - Prob. 51ECh. 12.5 - Prob. 52ECh. 12.5 - 5156 Determine whether the planes are parallel,...Ch. 12.5 - 5156 Determine whether the planes are parallel,...Ch. 12.5 - 5156 Determine whether the planes are parallel,...Ch. 12.5 - 5156 Determine whether the planes are parallel,...Ch. 12.5 - Prob. 57ECh. 12.5 - Prob. 58ECh. 12.5 - Prob. 59ECh. 12.5 - 5960 Find symmetric equations for the line of...Ch. 12.5 - Prob. 61ECh. 12.5 - Prob. 62ECh. 12.5 - Prob. 63ECh. 12.5 - Prob. 64ECh. 12.5 - Prob. 65ECh. 12.5 - Prob. 66ECh. 12.5 - Prob. 67ECh. 12.5 - Prob. 68ECh. 12.5 - Prob. 69ECh. 12.5 - Prob. 70ECh. 12.5 - 7172 Find the distance from the point to the given...Ch. 12.5 - Prob. 72ECh. 12.5 - Prob. 73ECh. 12.5 - Prob. 74ECh. 12.5 - Prob. 75ECh. 12.5 - Prob. 76ECh. 12.5 - Show that the lines with symmetric equations x=y=z...Ch. 12.5 - Prob. 78ECh. 12.5 - Prob. 79ECh. 12.5 - Prob. 80ECh. 12.5 - Prob. 81ECh. 12.5 - Prob. 82ECh. 12.5 - If a, b, and c are not all 0, show that the...Ch. 12.6 - a What does the equation y=x2 represent as a curve...Ch. 12.6 - a Sketch the graph of y=ex as a curve in 2. b...Ch. 12.6 - 38 Describe and sketch the surface. x2+z2=1Ch. 12.6 - 38 Describe and sketch the surface. 4x2+y2=4Ch. 12.6 - 38 Describe and sketch the surface. z=1y2Ch. 12.6 - 38 Describe and sketch the surface. y=z2Ch. 12.6 - 38 Describe and sketch the surface. xy=1Ch. 12.6 - 38 Describe and sketch the surface. z=sinyCh. 12.6 - a Find and identify the traces of the quadric...Ch. 12.6 - a Find and identify the traces of the quadric...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - Prob. 24ECh. 12.6 - Prob. 25ECh. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - Prob. 28ECh. 12.6 - 2930 Sketch and identify a quadric surface that...Ch. 12.6 - Prob. 30ECh. 12.6 - Prob. 31ECh. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - Prob. 33ECh. 12.6 - Prob. 34ECh. 12.6 - Prob. 35ECh. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - Prob. 37ECh. 12.6 - Prob. 38ECh. 12.6 - Prob. 39ECh. 12.6 - Prob. 40ECh. 12.6 - Prob. 41ECh. 12.6 - Prob. 42ECh. 12.6 - Prob. 43ECh. 12.6 - Prob. 44ECh. 12.6 - Find an equation for the surface obtained by...Ch. 12.6 - Prob. 46ECh. 12.6 - Prob. 47ECh. 12.6 - Find an equation for the surface consisting of all...Ch. 12.6 - Traditionally, the earths surface has been modeled...Ch. 12.6 - A cooling tower for a nuclear reactor is to be...Ch. 12.6 - Show that if the point (a,b,c) lies on the...Ch. 12.6 - Show that the curve of intersection of the...Ch. 12.6 - Prob. 53ECh. 12.R - What is the difference between a vector and a...Ch. 12.R - Prob. 2CCCh. 12.R - Prob. 3CCCh. 12.R - Prob. 4CCCh. 12.R - Prob. 5CCCh. 12.R - Prob. 6CCCh. 12.R - Prob. 7CCCh. 12.R - Prob. 8CCCh. 12.R - Prob. 9CCCh. 12.R - Prob. 10CCCh. 12.R - Prob. 11CCCh. 12.R - Prob. 12CCCh. 12.R - Prob. 13CCCh. 12.R - Prob. 14CCCh. 12.R - Prob. 15CCCh. 12.R - Prob. 16CCCh. 12.R - Prob. 17CCCh. 12.R - What are the traces of a surface? How do you find...Ch. 12.R - Prob. 19CCCh. 12.R - Prob. 1TFQCh. 12.R - Prob. 2TFQCh. 12.R - Prob. 3TFQCh. 12.R - Prob. 4TFQCh. 12.R - Prob. 5TFQCh. 12.R - Prob. 6TFQCh. 12.R - Prob. 7TFQCh. 12.R - Prob. 8TFQCh. 12.R - Prob. 9TFQCh. 12.R - Prob. 10TFQCh. 12.R - Prob. 11TFQCh. 12.R - Prob. 12TFQCh. 12.R - Prob. 13TFQCh. 12.R - Prob. 14TFQCh. 12.R - Prob. 15TFQCh. 12.R - Prob. 16TFQCh. 12.R - Prob. 17TFQCh. 12.R - Prob. 18TFQCh. 12.R - Prob. 19TFQCh. 12.R - Prob. 20TFQCh. 12.R - Prob. 21TFQCh. 12.R - Prob. 22TFQCh. 12.R - Prob. 1ECh. 12.R - Prob. 2ECh. 12.R - Prob. 3ECh. 12.R - Calculate the given quantity if...Ch. 12.R - Prob. 5ECh. 12.R - Prob. 6ECh. 12.R - Prob. 7ECh. 12.R - Prob. 8ECh. 12.R - Prob. 9ECh. 12.R - Prob. 10ECh. 12.R - Prob. 11ECh. 12.R - Prob. 12ECh. 12.R - Prob. 13ECh. 12.R - Prob. 14ECh. 12.R - Prob. 15ECh. 12.R - Prob. 16ECh. 12.R - Prob. 17ECh. 12.R - Prob. 18ECh. 12.R - Prob. 19ECh. 12.R - Prob. 20ECh. 12.R - Find the point in which the line with parametric...Ch. 12.R - Prob. 22ECh. 12.R - Prob. 23ECh. 12.R - a Show that the planes x+yz=1 and 2x3y+4z=5 are...Ch. 12.R - Prob. 25ECh. 12.R - a Find an equation of the plane that passes...Ch. 12.R - Prob. 27ECh. 12.R - Prob. 28ECh. 12.R - Prob. 29ECh. 12.R - Prob. 30ECh. 12.R - Prob. 31ECh. 12.R - Prob. 32ECh. 12.R - Prob. 33ECh. 12.R - Prob. 34ECh. 12.R - Prob. 35ECh. 12.R - Prob. 36ECh. 12.R - Prob. 37ECh. 12.R - Prob. 38ECh. 12.P - Each edge of a cubical box has length 1 m. The box...Ch. 12.P - Let B be a solid box with length L, width W, and...Ch. 12.P - Let L be the line of intersection of the planes...Ch. 12.P - A plane is capable of flying at a speed of 180...Ch. 12.P - Prob. 5PCh. 12.P - Find an equation of the largest sphere that passes...Ch. 12.P - Prob. 7PCh. 12.P - A solid has the following properties. When...
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