Lines and Planes in Space

1. Find equations of the line passing through the point \( (-3, 1, 5)\)  with direction vector \({\bf v} = \langle 1, -1, 2 \rangle\).    solution

2. Find  parametric and symmetric equations for the line passing through the points \( (-3, 5, -1)\) and \((4, -1, 2)\).    solution

3. A point \(\displaystyle{P(1,-2,3)}\) and direction vector \({\bf v}=\langle 1,2,4\rangle\) are given. Find the point of intersection of the line with the xy-plane.    solution

4. Find the distance between point \( M = (2, -1, 5)\)  and the line \(\displaystyle{x-2 = \frac{y-3}{3}=\frac{z+1}{2}}\).  solution

5. Determine whether the following lines are equal, parallel but not equal, skew or intersecting.

 (a)  \(\displaystyle{L_1: x=2t, y=1-t, z=3-3t, \quad t \in \mathbb{R} } \)  and  \(\displaystyle{L_2: \frac{2-x}{4}=\frac{y+2}{2}=\frac{z-1}{6}  } \)    solution

(b)  \(\displaystyle{L_1: x=1+t, y=-1+3t, z=4-t, \quad t \in \mathbb{R} } \)  and  \( \displaystyle{L_2: x=2s, y=1+s, z=-3+4s, \quad s \in \mathbb{R} } \)    solution

6. Find the general equation of the plane that passes through point \(P\) and has normal vector \({\bf n}\).    \( \quad \displaystyle {P(5, 7, -2),  {\bf n}=\langle 1, -2,3 \rangle }\)    solution

7. Find an equation of the plane that passes through point \( (1, -2, 3)\) and contains the line given by \( \quad \displaystyle x-1=\frac{y}{2}=\frac{z+1}{5}\).    solution

8. Find parametric equations of the line passing through the point \(P(-2,1,3)\) that is perpendicular to the plane whose equation is \(2x+3y-5z=8\).   solution

9. Find the point at which the line with parametric equations \( x=2+3t, \, y=-4t, \, z=5t \) intersects the plane \(4(x-2)+5(y-2)-2z=0 \).   solution

10. Find an equation for the plane which passes through the points \((-1,2,3), \, (1,1,1)\), and \((0,-1,0)\).   solution

11. Find the general form for the equation of the plane consisting of all points which are equidistant (i.e. the same distance) from the points  (-2, 0, 4)  and  (4, 6, 0).   solution

12. Find an equation of the plane that goes through point \(P(-4,2,1)\) that also contains the line \( x=2-5t, y=1+3t, z=-2+t\).   solution

13. Find an equation of the plane that passes through point \(P(-1,2,1)\) and is perpendicular to the line of intersection of the planes given by \(x+y-z-2=0\) and \(2x-y+3z-1=0\).  solution alt solution

14. Find the distance between point \( P = (2, -1, 5)\)  and the plane given by \( 3x − y + z = 4\).    solution

15. Determine if the planes are parallel, orthogonal, or neither.  Find the angle between them if they are not parallel or orthogonal.   solution

(a) \(x+y-2z=0\) and \(2x+2y-4z=10\)     (b) \(2x+y-3z=2\) and \(4x-2y+2z=5\)   (c)  \(x+y+z=3\) and \(x-2y+3z=1\)