Vector Planes (DP IB Maths: AA HL)

The points A(2, 1, 0), B(-1, 4, 1) and C(1, 0, 3) lie on a plane .

Find an equation for in the form  where

Determine whether the point D(-2, 2, 5) lies on .

The plane has equation .

The line has equation . 

The plane  and the line  intersect at the point X.

Find the coordinates of X.

Find the acute angle, in degrees, between the line  and the plane .

The point P(1,-3, 1) lies on the line .

Find the exact value of PX.

Hence find the shortest distance between the point P and the plane .

Find the acute angle, in radians, between the two planes and  which can be defined by the equations:

.

The line L given by the Cartesian equation  lies on the plane The point P(4, 0, -3)  also lies on

Show that the vectors  and  are parallel to

Hence find the Cartesian equation of .

Consider the plane  defined by the Cartesian equation and the line  defined by the vector equation .

Show that the line  is parallel to the plane  but does not lie in the plane.

The line  is perpendicular to the plane and passes through the point P(7, -4, 9) .

Find a vector equation of the line .

Find the coordinates of the point where the line  and the planeintersect.

Hence find the shortest distance between the line  and the plane .

Consider the two planes defined by the Cartesian equations:

The line is the intersection of the planes  and .

Show that the line  is parallel to the vector .

The point P lies on both planes.

(i)

Find the values of and .

(ii)

Hence write down a vector equation of the line .

A third plane has the Cartesian equation .

Use algebra to show that the three planes intersect at a unique point Q and find the coordinates of Q.

Consider the three planes with Cartesian equations:

where  is a real constant. 

In the case when the three planes do not intersect at a unique point, find the value of   and state the geometrical relationship between the three planes.

In the case when  find the coordinates of the point of intersection between the three planes.

Two parallel planes are defined by the equations:

Show that  and find the value of .

Write down a vector equation of the line  that is perpendicular to both planes and goes through the point P(11, -3, 5).

Find the coordinates of the point where the line  intersects the plane

Hence find the shortest distance between the two planes and ${\Pi }_2$.

The plane has the vector equation 

Find a vector that is perpendicular to the plane .

Q is the point on the plane   that is closest to the point P(4, 0, -3). Find the coordinates of the point Q.

Hence find the reflection of the point P in the plane

Two planes are defined by the Cartesian equations:

Find the acute angle, in radians, between and .

A third plane  is defined by the equation  where .

The plane is perpendicular to the plane . Find the value of .

Determine whether the points A(1, -1, 8) , B(0, 10, 15) , C (-2, -6. 10) and D(3, -5, 3) can lie in the same plane.

The plane  has vector equation  

The line has vector equation  

The plane  and the line intersect at the point . 

Find the coordinates of .

Find the acute angle, in degrees, between the line  and the plane .

The point P(2, 6, -1) lies on the line .

Find the shortest distance between the point P and the plane . Fully justify your answer.

Find the acute angle, in radians, between the two planes and  which can be defined by the equations:

.

The plane is defined by the equation and the line is defined by the vector equation  .

Show that the line lies on the plane

The plane is defined by the equation ,

Show that the plane   is parallel to the plane .

Find a vector equation of the line that is perpendicular to both planes and passes through the point P(3, 1, 4).

Hence find the shortest distance between  and .

The plane has the Cartesian equation . 

The line has the Cartesian equation  where

Show that the is not parallel to the plane .

Given that the acute angle between the line  and the plane  is 60°, find the possible values of .

Consider the two planes defined by the Cartesian equations:

The line  is the intersection of the planes  

Find a vector equation of the line . Give your answer in the form where .

A third plane  has the Cartesian equation  where . The three planes do not meet at a unique point.

Find the exact value of  and determine the geometrical relationship between the three planes.

Consider the four planes with Cartesian equations:

where  and  are real constants.

In the case where there is no unique point of intersection of the three planes and , find the value of and give a geometric interpretation of the three planes.

In the case where , find the coordinates of the point of intersection between the three planes

In the case where there is a common line of intersection between the three planes , find the values of  and .

The point P(2, 0, -1)  is reflected in the plane which has equation 

Find the coordinates of the reflection of P in the plane .

The line  passes through the point P and intersects the plane at the point Q(8, 3, 11) . The line is reflected in the plane  to form line .

Find a vector equation of the line .

Find the acute angle, in degrees, between the lines and .

Two planes are defined by the equations:

Find the exact value of  where  is the acute angle between  and .

 and  intersect at the line . A third plane is defined by the equation  where  and  is perpendicular to .  When  the line lies on all three planes.

Find the values of and .

Given that  and  intersect at the line  intersect at the line The shortest distance between the lines  is .

Find the shortest distance between the lines and . Give your answer as an exact value.

The plane is defined by the Cartesian equation . 

The line  is defined by the Cartesian equation . 

Determine whether the point P(5, 8, 15) is closer to the plane  or the line .

Fully justify your answer.