<< Chapter < Page Chapter >> Page >

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

Got questions? Get instant answers now!

Show that line x 1 2 = y + 1 3 = z 2 4 is parallel to plane x 2 y + z = 6 .

Got questions? Get instant answers now!

Find the real number α such that the line of parametric equations x = t , y = 2 t , z = 3 + t , t is parallel to the plane of equation α x + 5 y + z 10 = 0.

Got questions? Get instant answers now!

For the following exercises, points P , Q , and R are given.

  1. Find the general equation of the plane passing through P , Q , and R .
  2. Write the vector equation n · P S = 0 of the plane at a., where S ( x , y , z ) is an arbitrary point of the plane.
  3. Find parametric equations of the line passing through the origin that is perpendicular to the plane passing through P , Q , and R .

P ( 1 , 1 , 1 ) , Q ( 2 , 4 , 3 ) , and R ( −1 , −2 , −1 )

a. −2 y + 3 z 1 = 0 ; b. 0 , −2 , 3 · x 1 , y 1 , z 1 = 0 ; c. x = 0 , y = −2 t , z = 3 t , t

Got questions? Get instant answers now!

P ( −2 , 1 , 4 ) , Q ( 3 , 1 , 3 ) , and R ( −2 , 1 , 0 )

Got questions? Get instant answers now!

Consider the planes of equations x + y + z = 1 and x + z = 0 .

  1. Show that the planes intersect.
  2. Find symmetric equations of the line passing through point P ( 1 , 4 , 6 ) that is parallel to the line of intersection of the planes.

a. Answers may vary; b. x 1 1 = z 6 −1 , y = 4

Got questions? Get instant answers now!

Consider the planes of equations y + z 2 = 0 and x y = 0 .

  1. Show that the planes intersect.
  2. Find parametric equations of the line passing through point P ( −8 , 0 , 2 ) that is parallel to the line of intersection of the planes.
Got questions? Get instant answers now!

Find the scalar equation of the plane that passes through point P ( −1 , 2 , 1 ) and is perpendicular to the line of intersection of planes x + y z 2 = 0 and 2 x y + 3 z 1 = 0 .

2 x 5 y 3 z + 15 = 0

Got questions? Get instant answers now!

Find the general equation of the plane that passes through the origin and is perpendicular to the line of intersection of planes x + y + 2 = 0 and z 3 = 0 .

Got questions? Get instant answers now!

Determine whether the line of parametric equations x = 1 + 2 t , y = −2 t , z = 2 + t , t intersects the plane with equation 3 x + 4 y + 6 z 7 = 0 . If it does intersect, find the point of intersection.

The line intersects the plane at point P ( −3 , 4 , 0 ) .

Got questions? Get instant answers now!

Determine whether the line of parametric equations x = 5 , y = 4 t , z = 2 t , t intersects the plane with equation 2 x y + z = 5 . If it does intersect, find the point of intersection.

Got questions? Get instant answers now!

Find the distance from point P ( 1 , 5 , −4 ) to the plane of equation 3 x y + 2 z 6 = 0 .

16 14

Got questions? Get instant answers now!

Find the distance from point P ( 1 , −2 , 3 ) to the plane of equation ( x 3 ) + 2 ( y + 1 ) 4 z = 0 .

Got questions? Get instant answers now!

For the following exercises, the equations of two planes are given.

  1. Determine whether the planes are parallel, orthogonal, or neither.
  2. 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.

[T] x + y + z = 0 , 2 x y + z 7 = 0

a. The planes are neither parallel nor orthogonal; b. 62 °

Got questions? Get instant answers now!

5 x 3 y + z = 4 , x + 4 y + 7 z = 1

Got questions? Get instant answers now!

x 5 y z = 1 , 5 x 25 y 5 z = −3

a. The planes are parallel.

Got questions? Get instant answers now!

[T] x 3 y + 6 z = 4 , 5 x + y z = 4

Got questions? Get instant answers now!

Show that the lines of equations x = t , y = 1 + t , z = 2 + t , t , and x 2 = y 1 3 = z 3 are skew, and find the distance between them.

1 6

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

  1. Find the radius of the sphere with center C tangent to the given plane.
  2. Find point P of tangency.

a. 18 29 ; b. P ( 51 29 , 130 29 , 62 29 )

Got questions? Get instant answers now!

Consider the plane of equation x y z 8 = 0 .

  1. Find the equation of the sphere with center C at the origin that is tangent to the given plane.
  2. Find parametric equations of the line passing through the origin and the point of tangency.
Got questions? Get instant answers now!

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.

This figure is the image of two children throwing a ball. The path of the ball is represented with an arc. The distance from the child throwing the ball to the point where the ball hits is 5 feet. The distance from the second child to where the ball hits is 3 feet.

4 x 3 y = 0

Got questions? Get instant answers now!

[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 a x + b y + c z = 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 { ( x , y , z ) | a x + b y + c z d , x 0 , y 0 , z 0 } , and the budget plane is the part of the plane of equation a x + b y + c z = d for which x 0 , y 0 , and z 0 . Consider a = $ 8 , b = $ 5 , c = $ 10 , and d = $ 500 .

  1. Use a CAS to graph the budget set and budget plane.
  2. For z = 25 , find the new budget equation and graph the budget set in the same system of coordinates.
Got questions? Get instant answers now!

[T] Consider r ( t ) = sin t , cos t , 2 t the position vector of a particle at time t [ 0 , 3 ] , where the components of r are expressed in centimeters and time is measured in seconds. Let O P be the position vector of the particle after 1 sec.

  1. Determine the velocity vector v ( 1 ) of the particle after 1 sec.
  2. 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 .
  3. Use a CAS to visualize the path of the particle along with the velocity vector and normal plane at point P .

a. v ( 1 ) = cos 1 , sin 1 , 2 ; b. ( cos 1 ) ( x sin 1 ) ( sin 1 ) ( y cos 1 ) + 2 ( z 2 ) = 0 ;
c.
This figure is the first octant of the 3-dimensional coordinate system. It has a parallelogram grid drawn representing a plane. There is a curve from y = 1 increasing. The curve intersects the plane. At the point the curve intersects the plane, there is a vector labeled “v(1).” It is upward parallel to the z-axis.

Got questions? Get instant answers now!

[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).

This figure is a picture of a rectangular solar grid on a roof. The corners of the rectangle are labeled A, B, C, D. There are two vectors, the first is from A to D. The second is from A to B.
  1. 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 A B × A D .
  2. Find parametric equations of line L 1 that passes through the center of the solar panel and has direction vector s = 1 3 i + 1 3 j + 1 3 k , which points toward the position of the Sun at a particular time of day.
  3. Find symmetric equations of line L 2 that passes through the center of the solar panel and is perpendicular to it.
  4. Determine the angle of elevation of the Sun above the solar panel by using the angle between lines L 1 and L 2 .
Got questions? Get instant answers now!
Practice Key Terms 9

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?

Ask