# Exercises for Equations of Lines and Planes in Space

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**In exercises 1 - 4, points \( P\) and \( Q\) are given. Let \( L\) be the line passing through points \( P\) and \( Q\).**

**a. Find the vector equation of line \( L\).**

**b. Find parametric equations of line \( L\).**

**c. Find symmetric equations of line \( L\).**

**d. Find parametric equations of the line segment determined by \( P\) and \( Q\).**

1) \( P(−3,5,9), \quad Q(4,−7,2)\)

**Answer:**- a. \(\vecs r=⟨−3,5,9⟩+t⟨7,−12,−7⟩, \quad t∈R;\)

b. \( x=−3+7t, \quad y=5−12t, \quad z=9−7t, \quad t∈R;\)

c. \(\dfrac{x+3}{7}=\dfrac{y−5}{−12}=\dfrac{z−9}{−7};\)

d. \(x=−3+7t, \quad y=5−12t, \quad z=9−7t, \quad 0 \le t \le 1\)

2) \( P(4,0,5), \quad Q(2,3,1)\)

3) \( P(−1,0,5), \quad Q(4,0,3)\)

**Answer:**- a. \(\vecs r=⟨−1,0,5⟩+t⟨5,0,−2⟩, \quad t∈R;\)

b. \( x=−1+5t,y=0,z=5−2t, \quad t∈R;\)

c. \(\dfrac{x+1}{5}=\dfrac{z−5}{−2}, \quad y=0;\)

d. \(x=−1+5t, \quad y=0, \quad z=5−2t, \quad t∈[0,1]\)

4) \( P(7,−2,6), \quad Q(−3,0,6)\)

**For exercises 5 - 8, point \( P\) and vector \(\vecs v\) are given. Let \( L\) be the line passing through point \( P\) with direction \(\vecs v\).**

**a. Find parametric equations of line \( L\).**

**b. Find symmetric equations of line \( L\).**

**c. Find the intersection of the line with the \(xy\)-plane.**

5) \( P(1,−2,3),\,\vecs v=⟨1,2,3⟩\)

**Answer:**- a. \(x=1+t, \quad y=−2+2t, \quad z=3+3t, \quad t∈R;\)

b. \( \dfrac{x−1}{1}=\dfrac{y+2}{2}=\dfrac{z−3}{3};\)

c. \((0,−4,0)\)

6) \( P(3,1,5), \,\vecs v=⟨1,1,1⟩\)

7) \( P(3,1,5), \,\vecs v=\vecd{QR},\) where \( Q(2,2,3)\) and \( R(3,2,3)\)

**Answer:**- a. \(x=3+t, \quad y=1, \quad z=5, \quad t∈R;\)

b. \( y=1, \quad z=5;\)

c. The line does not intersect the \(xy\)-plane.

8) \( P(2,3,0), \,\vecs v=\vecd{QR},\) where \( Q(0,4,5)\) and \( R(0,4,6)\)

**For exercises 9 and 10, line \( L\) is given.**

**a. Find a point \( P\) that belongs to the line and a direction vector \(\vecs v\) of the line. Express \(\vecs v\) in component form.**

**b. Find the distance from the origin to line \( L\).**

9) \( x=1+t, \quad y=3+t, \quad z=5+4t, \quad t∈R\)

**Answer:**- a. A possible point and direction vector are \(P(1,3,5)\) and \(\vecs v=⟨1,1,4⟩\), but these answers are not unique.

b. \( \sqrt{3} \) units

10) \( −x=y+1, \quad z=2\)

11) Find the distance between point \( A(−3,1,1)\) and the line of symmetric equations

\( x=−y=−z.\)

**Answer:**- \( \frac{2\sqrt{2}}{\sqrt{3}} = \frac{2\sqrt{6}}{3} \) units

12) Find the distance between point \( A(4,2,5)\) and the line of parametric equations

\( x=−1−t, \; y=−t, \; z=2, \; t∈R.\)

**For exercises 13 - 14, lines \( L_1\) and \( L_2\) are given.**

**a. Verify whether lines \( L_1\) and \( L_2\) are parallel.**

**b. If the lines \( L_1\) and \( L_2\) are parallel, then find the distance between them.**

13) \( L_1:x=1+t, \quad y=t, \quad z=2+t, \quad t∈R\) and \(L_2:x−3=y−1=z−3\)

**Answer:**- a. Parallel;

b. \( \frac{\sqrt{2}}{\sqrt{3}} = \frac{\sqrt{6}}{3} \) units

14) \( L_1:x=2, \quad y=1, \quad z=t, \quad t∈R\) and \( L_2:x=1, \quad y=1, \quad z=2−3t, \quad t∈R\)

15) Show that the line passing through points \( P(3,1,0)\) and \( Q(1,4,−3)\) is perpendicular to the line with equation \( x=3t, \quad y=-32+8t, \quad z=−9+6t, \quad t∈R.\)

**Answer:**- \( \vecd{PQ} = \langle -2, 3, -3 \rangle\) is the direction vector of the line through points \(P\) and \(Q\), and the direction vector of the line defined by the parametric equations above is \(\vecs v = \langle 3, 8, 6 \rangle.\)

Since \(\vecs v \cdot \vecd{PQ} = -6 + 24 - 18 = 0\), the two direction vectors are orthogonal.

Now all we need to show is that the two lines intersect.

The line through points \( P(3,1,0)\) and \( Q(1,4,−3)\) has parametric equations: \(x = 3 - 2u\), \(y = 1 + 3u\), and \(z = -3u\).

Setting the \(x\)- and \(z\)-coordinates of the two lines equal, we obtain the system of equations:

\[3t = 3 - 2u \quad\text{and}\quad -9 + 6t = -3u \nonumber\]

Solving this system using substitution gives us, \(u = -3\) and \(t = 3\). Plugging these values of \(t\) and \(u\) back into the parametric equations of these two lines gives us the intersection point with coordinates \(\left(9, -8, 9\right)\) on both lines.

Therefore the lines intersect and the line through points \(P\) and \(Q\) with direction vector \(\vecd{PQ} \) is perpendicular to the other line.

16) Are the lines of equations \( x=−2+2t, \quad y=−6, \quad z=2+6t, \quad t∈R\) and \( x=−1+t, \quad y=1+t, \quad z=t, \quad t∈R,\) perpendicular to each other?

17) Find the point of intersection of the lines of equations \( x=−2y=3z\) and \( x=−5−t, \quad y=−1+t, \quad z=t−11, \quad t∈R.\)

**Answer:**- \( (−12,6,−4)\)

18) Find the intersection point of the \(x\)-axis with the line of parametric equations \( x=10+t, \quad y=2−2t, \quad z=−3+3t, \quad t∈R.\)

**For exercises 19 - 22, lines \( L_1\) and \( L_2\) are given. Determine whether the lines are equal, parallel but not equal, skew, or intersecting.**

19) \( L_1:x=y−1=−z\) and \( L_2:x−2=−y=\dfrac{z}{2}\)

**Answer:**- The lines are skew.

20) \( L_1:x=2t, \quad y=0, \quad z=3, \quad t∈R\) and \( L_2:x=0, \quad y=8+s, \quad z=7+s, \quad s∈R\)

21) \( L_1:x=−1+2t, \quad y=1+3t, \quad z=7t, \quad t∈R\) and \( L_2:x−1=\frac{2}{3}(y−4)=\frac{2}{7}z−2\)

**Answer:**- The lines are equal.

22) \( L_1:3x=y+1=2z\) and \( L_2:x=6+2t, \quad y=17+6t, \quad z=9+3t, \quad t∈R\)

23) Consider line \( L\) of symmetric equations \( x−2=−y=\dfrac{z}{2}\) and point \( A(1,1,1).\)

a. Find parametric equations for a line parallel to \( L\) that passes through point \( A\).

b. Find symmetric equations of a line skew to \( L\) and that passes through point \( A\).

c. Find symmetric equations of a line that intersects \( L\) and passes through point \( A\).

**Answer:**- a. \(x=1+t, \quad y=1−t, \quad z=1+2t, \quad t∈R\)

b. For instance, the line passing through \( A\) with direction vector \( j:x=1,z=1\)

c. For instance, the line passing through \( A\) and point \( (2,0,0)\) that belongs to \( L\) is a line that intersects; \( L:\frac{x−1}{−1}=y−1=z−1\)

24) Consider line \( L\) of parametric equations \( x=t, \quad y=2t, \quad z=3, \quad t∈R.\)

a. Find parametric equations for a line parallel to \( L\) that passes through the origin.

b. Find parametric equations of a line skew to \( L\) that passes through the origin.

c. Find symmetric equations of a line that intersects \( L\) and passes through the origin.

**For exercises 25 - 28, point \( P\) and vector \(\vecs n\) are given.**

**a. Find the standard form of the equation of the plane that passes through \( P\) and has normal vector \(\vecs n\).**

**b. Find the general form of the equation of the plane that passes through \( P\) and has normal vector \(\vecs n\).**

25) \( P(0,0,0), \quad \vecs n=3\mathbf{\hat i}−2\mathbf{\hat j}+4\mathbf{\hat k}\)

**Answer:**- a. \(3x−2y+4z=0\)

b. \(3x−2y+4z=0\)

26) \( P(3,2,2), \quad \vecs n=2\mathbf{\hat i}+3\mathbf{\hat j}−\mathbf{\hat k}\)

27) \( P(1,2,3), \quad \vecs n=⟨1,2,3⟩\)

**Answer:**- a. \((x−1)+2(y−2)+3(z−3)=0\)

b. \(x+2y+3z−14=0\)

28) \( P(0,0,0), \quad \vecs n=⟨−3,2,−1⟩\)

**For exercises 29 - 32, the equation of a plane is given.**

**a. Find normal vector \(\vecs n\) to the plane. Express \(\vecs n\) using standard unit vectors.**

**b. Find the intersections of the plane with each of the coordinate axes (its intercepts).**

**c. Sketch the plane.**

29) [T] \( 4x+5y+10z−20=0\)

**Answer:**- a. \(\vecs n=4\mathbf{\hat i}+5\mathbf{\hat j}+10\mathbf{\hat k}\)

b. \((5,0,0), \,(0,4,0),\) and \( (0,0,2)\)c.

30) \( 3x+4y−12=0\)

31) \( 3x−2y+4z=0\)

**Answer:**- a. \(\vecs n=3\mathbf{\hat i}−2\mathbf{\hat j}+4\mathbf{\hat k}\)

b. \((0,0,0)\)c.

32) \( x+z=0\)

33) Given point \( P(1,2,3)\) and vector \(\vecs n=\mathbf{\hat i}+\mathbf{\hat j}\), find point \( Q\) on the \(x\)-axis such that \( \vecd{PQ}\) and \(\vecs n\) are orthogonal.

**Answer:**- \( (3,0,0)\)

34) Show there is no plane perpendicular to \(\vecs n=\mathbf{\hat i}+\mathbf{\hat j}\) that passes through points \( P(1,2,3)\) and \( Q(2,3,4)\).

35) Find parametric equations of the line passing through point \( P(−2,1,3)\) that is perpendicular to the plane of equation \( 2x−3y+z=7.\)

**Answer:**- \( x=−2+2t, \quad y=1−3t, \quad z=3+t, \quad t∈R\)

36) Find symmetric equations of the line passing through point \( P(2,5,4)\) that is perpendicular to the plane of equation \( 2x+3y−5z=0.\)

37) Show that line \( \dfrac{x−1}{2}=\dfrac{y+1}{3}=\dfrac{z−2}{4}\) is parallel to plane \( x−2y+z=6\).

38) Find the real number \( α\) such that the line of parametric equations \( x=t, \quad y=2−t, \quad z=3+t, \quad t∈R\) is parallel to the plane of equation \( αx+5y+z−10=0.\)

**For exercises 39 - 42, the equations of two planes are given.**

**a. Determine whether the planes are parallel, orthogonal, or neither.**

**b. If the planes are neither parallel nor orthogonal, then find the measure of the angle between the planes. Express the answer in degrees rounded to the nearest integer.**

**c. If the planes intersect, find the line of intersection of the planes, providing the parametric equations of this line.**

39) [T] \( x+y+z=0, \quad 2x−y+z−7=0\)

**Answer:**- a. The planes are neither parallel nor orthogonal.

b. \(62°\)

c. \(x = -1 + 2t\)

\(y = -4 + t\)

\(z = 5 - 3t\)

40) \( 5x−3y+z=4, \quad x+4y+7z=1\)

41) \( x−5y−z=1, \quad 5x−25y−5z=−3\)

**Answer:**- a. The planes are parallel.

42) [T] \( x−3y+6z=4, \quad 5x+y−z=4\)

**For exercises 43 - 46, determine whether the given line intersects with the given plane. If they do intersect, state the point of intersection.**

43) Plane: \(2x + y - z = 11\) Line: \(x = 1 + t, \quad y = 3 - 2t, \quad z = 2 +4t\)

**Answer:**- They intersect at point \( (-1, 7, -6) \).

44) Plane: \(-x + 2y + z = 2\) Line: \(x = 1 + 2t, \quad y = -2 + t, \quad z = 5 - 3t\)

**Answer:**- They intersect at point \( \left(-\frac{1}{3}, \, -\frac{8}{3}, \, 7\right) \).

45) Plane: \(x - 3y + 2z = 4\) Line: \(x = 2 - t, \quad y = t, \quad z = 4 +2t\)

**Answer:**- The line does not intersect with this plane.

46) Plane: \(x - 3y + 2z = 10\) Line: \(x = 2 - t, \quad y = t, \quad z = 4 +2t\)

**Answer:**- The line is actually fully contained in this plane, so every point on the line is on the plane. For example, when \(t = 0\) we have the point, \((2, 0, 4)\).

47) Show that the lines of equations \( x=t, \quad y=1+t, \quad z=2+t, \quad t∈R,\) and \( \dfrac{x}{2}=\dfrac{y−1}{3}=z−3\) are skew, and find the distance between them.

**Answer:**- \( \frac{1}{\sqrt{6}} = \frac{\sqrt{6}}{6}\) units

48) Show that the lines of equations \( x=−1+t, \quad y=−2+t, \quad z=3t, \quad t∈R,\) and \( x=5+s, \quad y=−8+2s, \quad z=7s, \quad s∈R\) are skew, and find the distance between them.

49) Consider point \( C(−3,2,4)\) and the plane of equation \( 2x+4y−3z=8\).

a. Find the radius of the sphere with center \(C\) tangent to the given plane.

b. Find point \(P\) of tangency.

**Answer:**- a. \(r = \frac{18}{\sqrt{29}} = \frac{18\sqrt{29}}{29}\)

b. \(P\left(−\frac{51}{29},\frac{130}{29},\frac{62}{29}\right)\)

50) Consider the plane of equation \( x−y−z−8=0.\)

a. Find the equation of the sphere with center \(C\) at the origin that is tangent to the given plane.

b. Find parametric equations of the line passing through the origin and the point of tangency.

51) Two children are playing with a ball. The girl throws the ball to the boy. The ball travels in the air, curves \( 3\) ft to the right, and falls \( 5\) ft away from the girl (see the following figure). If the plane that contains the trajectory of the ball is perpendicular to the ground, find its equation.

**Answer:**- \( 4x−3y=0\)

52) [T] John allocates \( d\) dollars to consume monthly three goods of prices \( a,b\), and \( c\). In this context, the budget equation is defined as \( ax+by+cz=d,\) where \( x≥0,\, y≥0\), and \( z≥0\) represent the number of items bought from each of the goods. The budget set is given by \( \big\{(x,y,z)\,|\,ax+by+cz≤d,\;x≥0,\;y≥0,\;z≥0\big\},\) and the budget plane is the part of the plane of equation \( ax+by+cz=d\) for which \( x≥0,\,y≥0\), and \( z≥0\). Consider \( a=$8, \,b=$5, \,c=$10,\) and \( d=$500.\)

a. Use a CAS to graph the budget set and budget plane.

b. For \( z=25,\) find the new budget equation and graph the budget set in the same system of coordinates.

53) [T] Consider \(\vecs r(t)=⟨\sin t,\cos t,2t⟩\) the position vector of a particle at time \( t∈[0,3]\), where the components of \(\vecs r\) are expressed in centimeters and time is measured in seconds. Let \( \vecd{OP}\) be the position vector of the particle after \( 1\) sec.

a. Determine the velocity vector \(\vecs v(1)\) of the particle after \( 1\) sec.

b. Find the scalar equation of the plane that is perpendicular to \( v(1)\) and passes through point \( P\). This plane is called the normal plane to the path of the particle at point \( P\).

c. Use a CAS to visualize the path of the particle along with the velocity vector and normal plane at point \( P\).

**Answer:**- a. \(\vecs v(1)=⟨\cos 1,−\sin 1, 2⟩\)

b. \( (\cos 1)(x−\sin 1)−(\sin 1)(y−\cos 1)+2(z−2)=0\)

c.

54) [T] A solar panel is mounted on the roof of a house. The panel may be regarded as positioned at the points of coordinates (in meters) \( A(8,0,0), \, B(8,18,0), \, C(0,18,8),\) and \( D(0,0,8)\) (see the following figure).

a. Find the general form of the equation of the plane that contains the solar panel by using points \( A, \, B,\) and \( C\), and show that its normal vector is equivalent to \( \vecd{AB}×\vecd{AD}.\)

b. Find parametric equations of line \( L_1\) that passes through the center of the solar panel and has direction vector \(\vecs s=\frac{1}{\sqrt{3}}\mathbf{\hat i}+\frac{1}{\sqrt{3}}\mathbf{\hat j}+\frac{1}{\sqrt{3}}\mathbf{\hat k},\) which points toward the position of the Sun at a particular time of day.

c. Find symmetric equations of line \( L_2\) that passes through the center of the solar panel and is perpendicular to it.

d. Determine the angle of elevation of the Sun above the solar panel by using the angle between lines \( L_1\) and \( L_2\).

## Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

Problems 15 and 43 - 46 created by Paul Seeburger