The xy-coordinate plane, for example, is the plane where the z-coordinate of every point is zero. The scalar equation of the xy-plane is z⫽0. By setting zequal to zero in the equation of a plane, we are singling out those points in the plane that lie in the xy-coordinate plane. These are exactly the points on the intercept line, and by setting z⫽0 we obtain the equation.
In Example 4, for instance, the plane intersects the xy-coordinate plane in the line 3x⫺8y⫺8(0)⫹24⫽0 or 3x⫺8y⫹24⫽0. In Example 5, the plane inter- sects the xy-coordinate plane in the line 3x⫹2y⫺18⫽0 (there is no variable z in the equation of this plane, so setting zequal to zero does not change the equation).
EXAMPLE 6 Sketch the plane 5x⫺2y⫽0.
Solution
Since D⫽0, the point (0, 0, 0) satisfies the equation of the plane. So this plane contains the origin. Consequently the x- and y-intercepts are both zero. The nor- mal to this plane is (5,⫺2, 0), so as with Example 5, this plane is parallel to the z-axis. But if the plane is parallel to the z-axis and contains the origin, it must contain the entire z-axis. You can reach the same conclusion by observing that every point (0, 0,z) on the z-axis satisfies the equation of the plane.
The set of planes with this property is illustrated in the given diagram.
Sketch the plane as a parallelogram, with the intersection line and the z-axis as sides. This parallelogram-shaped region repre- sents a section of the plane 5x⫺2y⫽0 in three dimensions.
From this set of planes, we choose the one which intercepts the xy-plane along the line with equation 5x⫺2y⫽0.
Part A
1. For each of the following, find the intersection of the line and the plane.
a. x⫽4⫺t,y⫽6⫹2t,z⫽ ⫺2⫹t and 2x⫺y⫹6z⫹10⫽0 b. x⫽3⫹4t,y⫽ ⫺2⫺6t,z⫽ᎏ1
2ᎏ⫺3t and 3x⫹4y⫺7z⫹7⫽0
Knowledge/
Understanding
Exercise 8.3
292
z
x ?
y
(5x – 2y = 0) z
x
y
C H A P T E R 8
c. x⫽5⫹t,y⫽4⫹2t,z⫽7⫹2t and 2x⫹3y⫺4z⫹7⫽0 d. r⫽(2, 14, 1)⫹t(⫺1,⫺1, 1) and 3x⫺y⫹2z⫹6⫽0 e. r⫽(5, 7, 3)⫹t(0, 1,⫺1) and z⫹5⫽0
2. a. Does the line r⫽(⫺2, 6, 5)⫹t(3, 2,⫺1) lie in the plane 3x⫺4y⫹z⫹25⫽0?
b. Does the line r⫽(4,⫺1, 2)⫹t(3, 2,⫺1) lie in the plane 3x⫺4y⫹z⫺17⫽0?
3. Where does the plane 3x⫺2y⫺7z⫺6⫽0 intersect
a. the x-axis? b. the y-axis? c. the z-axis?
Part B
4. a. In what point does the plane r⫽(6,⫺4, 3)⫹s(⫺2, 4, 7)⫹t(⫺7, 6,⫺3) intersect
i) the x-axis ii) the y-axis iii) the z-axis b. In what line does this plane intersect the
i) the xy-plane ii) the yz-plane iii) the xz-plane 5. Where does the line r⫽(6, 10, 1)⫹t(3, 4,⫺1) meet
a. the xy-plane b. the xz-plane c. the yz-plane 6. State whether it is possible for the lines and planes described below to
intersect in one point, in an infinite number of points, or in no points.
a. a line parallel to the x-axis and a plane perpendicular to the x-axis b. a line parallel to the y-axis and a plane parallel to the y-axis c. a line perpendicular to the z-axis and a plane parallel to the z-axis 7. Find the point of intersection of the plane 3x⫺2y⫹7z⫺31⫽0 with the
line that passes through the origin and is perpendicular to the plane.
8. Find the point at which the normal to the plane 4x⫺2y⫹5z⫹18⫽0 through the point (6,⫺2,⫺2) intersects the plane.
9. For each of the following planes, find the x-,y-, and z-intercepts and make a three-dimensional sketch.
a. 12x⫹3y⫹4z⫺12⫽0 b. x⫺2y⫺z⫺5⫽0 c. 2x⫺y⫹z⫹8⫽0 d. 4x⫺y⫹2z⫺16⫽0
Application Application Communication
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10. For each of the following planes, find the x-,y-, and z-intercepts, if they exist, and the intersections with the coordinate planes. Then make a three-dimen- sional sketch of the plane.
a. x⫹y⫺4⫽0 b. x⫺3⫽0 c. 2y⫹1⫽0
d. 3x⫹z⫺6⫽0 e. y⫺2z⫽0 f. x⫹y⫺z⫽0 Part C
11. For what values of kwill the line ᎏx⫺ 3
ᎏk ⫽ᎏy⫹ 2
ᎏ4 ⫽ᎏz⫹ 1
ᎏ6 intersect the plane x⫺4y⫹5z⫹5⫽0
a. in a single point?
b. in an infinite number of points?
c. in no points?
12. A plane has an x-intercept of a, a y-intercept of b, and a z-intercept of c, none of which is zero. Show that the equation of the plane is ᎏaxᎏ⫹ᎏb
yᎏ⫹ᎏc zᎏ⫽1.
Thinking/Inquiry/
Problem Solving
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