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8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781285740621

Chapter 12.5, Problem 22E

Textbook Problem

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**19–22** Determine whether the lines

To determine

Whether the lines

**1) Concept:**

i) Two lines are parallel if their corresponding direction vectors are parallel.

ii) For the lines to intersect, there must be one value of

iii) If lines are not parallel and intersecting then they must be skew.

**2) Given:**

**3) Calculation:**

The equations of lines are

The direction vector for the line

The direction vector for the line

Find

Since, corresponding direction vectors are not parallel, the lines aren’t parallel.

By using two non - zero vectors,

The lines aren’t parallel.

Comparing the line

Calculus (MindTap Course List)

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Two of the forces...Ch. 12.2 - Find the unit vectors that are parallel to the...Ch. 12.2 - a Find the unit vectors that are parallel to the...Ch. 12.2 - If A, B, and C are the vertices of a triangle,...Ch. 12.2 - Let C be the point on the line segment AB that is...Ch. 12.2 - a Draw the vectors a=3,2,b=2,1 and c=7,1. b Show,...Ch. 12.2 - Suppose that a and b are nonzero vectors that are...Ch. 12.2 - If r=x,y,z and r0=x0,y0,z0, describe the set of...Ch. 12.2 - If r=x,y,r1=x1,y1, and r2=x2,y2, describe the set...Ch. 12.2 - Figure 16 gives a geometric demonstration of...Ch. 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 - Suppose the three coordinate planes are all...Ch. 12.3 - Which of the following expressions are meaningful?...Ch. 12.3 - 210 Find ab. a=5,2, b=3,4Ch. 12.3 - 210 Find ab. a=1.5,0.4, b=4,6Ch. 12.3 - 210 Find ab. a=6,2,3, b=2,5,1Ch. 12.3 - 210 Find ab. a=4,1,14, b=6,3,8Ch. 12.3 - 210 Find ab. a=p,p,2p, b=2q,q,qCh. 12.3 - 210 Find ab. a=2i+j, b=i-j+kCh. 12.3 - 210 Find ab. a=3i+2j-k, b=4i+5kCh. 12.3 - 210 Find ab. |a|=7, |b|=4, the angle between a and...Ch. 12.3 - Find ab. |a|=80, |b|=50, the angle between a and b...Ch. 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 - a Show that ij=jki=0. b Show that ii=jj=kk=1.Ch. 12.3 - A street vendor sells a hamburgers, b hot dogs,...Ch. 12.3 - Find the angle between the vectors. First find an...Ch. 12.3 - Find the angle between the vectors. First find an...Ch. 12.3 - Find the angle between the vectors. First find an...Ch. 12.3 - Find the angle between the vectors. First find an...Ch. 12.3 - Find the angle between the vectors. First find an...Ch. 12.3 - Find the angle between the vectors. First find an...Ch. 12.3 - 2122 Find, correct to the nearest degree, the...Ch. 12.3 - 2122 Find, correct to the nearest degree, the...Ch. 12.3 - 2324 Determine whether the given vectors are...Ch. 12.3 - 2324 Determine whether the given vectors are...Ch. 12.3 - Use vectors to decide whether the triangle with...Ch. 12.3 - Find the values of x such that the angle between...Ch. 12.3 - Find a unit vector that is orthogonal to both i+j...Ch. 12.3 - Find two unit vectors that make an angle of 60...Ch. 12.3 - 2930 Find the acute angle between the lines....Ch. 12.3 - 2930 Find the acute angle between the lines....Ch. 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 - 3337 Find the direction cosines and direction...Ch. 12.3 - 3337 Find the direction cosines and direction...Ch. 12.3 - 3337 Find the direction cosines and direction...Ch. 12.3 - 3337 Find the direction cosines and direction...Ch. 12.3 - 3337 Find the direction cosines and direction...Ch. 12.3 - If a vector has direction angles =/4 and =/3, find...Ch. 12.3 - 3944 Find the scalar and vector projections of b...Ch. 12.3 - 3944 Find the scalar and vector projections of b...Ch. 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 - Find the scalar and vector projections of b onto...Ch. 12.3 - Find the scalar and vector projections of b onto...Ch. 12.3 - Show that the vector orthaab=bprojba is orthogonal...Ch. 12.3 - For the vectors in Exercise 40, find orthab and...Ch. 12.3 - If a=3,0,1, find a vector b such that compab=2.Ch. 12.3 - Suppose that a and b are nonzero vectors. a Under...Ch. 12.3 - Find the work done by a force F=8i6j+9k that moves...Ch. 12.3 - A tow truck drags a stalled car along a road. The...Ch. 12.3 - A sled is pulled along a level path through snow...Ch. 12.3 - A boat sails south with the help of a wind blowing...Ch. 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 - Find the angle between a diagonal of a cube and...Ch. 12.3 - Find the angle between a diagonal of a cube and a...Ch. 12.3 - A molecule of methane, CH4, is structured with the...Ch. 12.3 - If c=|a|b+|b|a, where a, b, and c are all nonzero...Ch. 12.3 - Prove Properties 2, 4, and 5 of the dot product...Ch. 12.3 - Suppose that all sides of a quadrilateral are...Ch. 12.3 - Use Theorem 3 to prove the Cauchy-Schwarz...Ch. 12.3 - The Triangle Inequality for vectors is...Ch. 12.3 - The Parallelogram Law states that...Ch. 12.3 - Show that if u+v and uv are orthogonal, then the...Ch. 12.3 - If is the angle between vectors a and b, show...Ch. 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 - 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 - 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 - 17 Find the cross product ab and verify that it is...Ch. 12.4 - If a=i2k and b=j+k, find ab. 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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 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 4144 Use intercepts to help sketch the plane....Ch. 12.5 - 4144 Use intercepts to help sketch the plane....Ch. 12.5 - 4144 Use intercepts to help sketch the plane....Ch. 12.5 - 4144 Use intercepts to help sketch the plane....Ch. 12.5 - 4547 Find the point at which the line intersects...Ch. 12.5 - 4547 Find the point at which the line intersects...Ch. 12.5 - 4547 Find the point at which the line intersects...Ch. 12.5 - Where does the line through (3,1,0) and (1,5,6)...Ch. 12.5 - Find direction numbers for the line of...Ch. 12.5 - Find the cosine of the angle between the planes...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 - 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 - 5758 a Find parametric equations for the line of...Ch. 12.5 - 5758 a Find parametric equations for the line of...Ch. 12.5 - 5960 Find symmetric equations for the line of...Ch. 12.5 - 5960 Find symmetric equations for the line of...Ch. 12.5 - Find an equation for the plane consisting of all...Ch. 12.5 - Find an equation for the plane consisting of all...Ch. 12.5 - Find an equation of the plane with x-intercept a,...Ch. 12.5 - a Find the point at which the given lines...Ch. 12.5 - Find parametric equations for the line through the...Ch. 12.5 - Find parametric equations for the line through the...Ch. 12.5 - Which of the following four planes are parallel?...Ch. 12.5 - Which of the following four lines are parallel?...Ch. 12.5 - 6970 Use the formula in Exercise 12.4.45 to find...Ch. 12.5 - 6970 Use the formula in Exercise 12.4.45 to find...Ch. 12.5 - 7172 Find the distance from the point to the given...Ch. 12.5 - 7172 Find the distance from the point to the given...Ch. 12.5 - 7374 Find the distance between the given parallel...Ch. 12.5 - 7374 Find the distance between the given parallel...Ch. 12.5 - Show that the distance between the parallel planes...Ch. 12.5 - Find equations of the planes that are parallel to...Ch. 12.5 - Show that the lines with symmetric equations x=y=z...Ch. 12.5 - Find the distance between the skew lines with...Ch. 12.5 - Let L1 be the line through the origin and the...Ch. 12.5 - Let L1 be the line through the points (1,2,6) and...Ch. 12.5 - Two tanks are participating in a battle...Ch. 12.5 - Give a geometric description of each family of...Ch. 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 - 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 - 2128 Match the equation with reasons for your...Ch. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - 2930 Sketch and identify a quadric surface that...Ch. 12.6 - 2930 Sketch and identify a quadric surface that...Ch. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - 3942 Use a computer with three-dimensional...Ch. 12.6 - 3942 Use a computer with three-dimensional...Ch. 12.6 - 3942 Use a computer with three-dimensional...Ch. 12.6 - 3942 Use a computer with three-dimensional...Ch. 12.6 - Sketch the region bounded by the surfaces z=x2+y2...Ch. 12.6 - Sketch the region bounded by the paraboloids...Ch. 12.6 - Find an equation for the surface obtained by...Ch. 12.6 - Find an equation for the surface obtained by...Ch. 12.6 - Find an equation for the surface consisting of all...Ch. 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 - Graph the surfaces z=x2+y2 and z=1y2 on a common...Ch. 12.R - What is the difference between a vector and a...Ch. 12.R - How do you add two vectors geometrically? How do...Ch. 12.R - If a is a vector and c is a scalar, how is ca...Ch. 12.R - How do you find the vector from one point to...Ch. 12.R - How do you find the dot product ab of two vectors...Ch. 12.R - How are dot products useful?Ch. 12.R - Write expressions for the scalar and vector...Ch. 12.R - How do you find the cross product ab of two...Ch. 12.R - How are cross products useful?Ch. 12.R - a How do you find the area of the parallelogram...Ch. 12.R - How do you find a vector perpendicular to a plane?Ch. 12.R - How do you find the angle between two intersecting...Ch. 12.R - Write a vector equation, parametric equations, and...Ch. 12.R - Write a vector equation and a scalar equation for...Ch. 12.R - a How do you tell if two vectors are parallel? b...Ch. 12.R - a Describe a method for determining whether three...Ch. 12.R - a How do you find the distance from a point to a...Ch. 12.R - What are the traces of a surface? How do you find...Ch. 12.R - Write equations in standard form of the six types...Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - a Find an equation of the sphere that passes...Ch. 12.R - Copy the vectors in the figure and use them to...Ch. 12.R - If u and v are the vectors shown in the figure,...Ch. 12.R - Calculate the given quantity if...Ch. 12.R - Find the values of x such that the vectors 3,2,x...Ch. 12.R - Find two unit vectors that are orthogonal to both...Ch. 12.R - Suppose that u(vw)=2. Find a (uv)w b u(wv) c v(uw)...Ch. 12.R - Show that if a, b, and c are in V3, then...Ch. 12.R - Find the acute angle between two diagonals of a...Ch. 12.R - Given the points...Ch. 12.R - a Find a vector perpendicular to the plane through...Ch. 12.R - A constant force F=3i+5j+10k moves an object along...Ch. 12.R - A boat is pulled onto shore using two ropes, as...Ch. 12.R - Find the magnitude of the torque about P if a 50-N...Ch. 12.R - 1517 Find parametric equations for the line. The...Ch. 12.R - 1517 Find parametric equations for the line. The...Ch. 12.R - 1517 Find parametric equations for the line. The...Ch. 12.R - 1820 Find an equation of the plane. The plane...Ch. 12.R - 1820 Find an equation of the plane. The plane...Ch. 12.R - 1820 Find an equation of the plane. The plane...Ch. 12.R - Find the point in which the line with parametric...Ch. 12.R - Find the distance from the origin to the line...Ch. 12.R - Determine whether the lines given by the symmetric...Ch. 12.R - a Show that the planes x+yz=1 and 2x3y+4z=5 are...Ch. 12.R - Find an equation of the plane through the fine of...Ch. 12.R - a Find an equation of the plane that passes...Ch. 12.R - Find the distance between the planes 3x+y4z=2 and...Ch. 12.R - 28-36. Identify and sketch the graph of each...Ch. 12.R - 28-36. Identify and sketch the graph of each...Ch. 12.R - 28-36. Identify and sketch the graph of each...Ch. 12.R - 28-36. Identify and sketch the graph of each...Ch. 12.R - 28-36. Identify and sketch the graph of each...Ch. 12.R - 28-36. Identify and sketch the graph of each...Ch. 12.R - 28-36. Identify and sketch the graph of each...Ch. 12.R - 28-36. Identify and sketch the graph of each...Ch. 12.R - 28-36. Identify and sketch the graph of each...Ch. 12.R - An ellipsoid is created by rotating the ellipse...Ch. 12.R - A surface consists of all points P such that the...Ch. 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 - Suppose v1 and v2 are vectors with,...Ch. 12.P - Find an equation of the largest sphere that passes...Ch. 12.P - Suppose a block of mass m is placed on an inclined...Ch. 12.P - A solid has the following properties. When...

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