Vector Planes (DP IB Maths: AA HL)

A plane has equation

A line with equation intersects at a point . 

Write down the equations of the line and the plane in their parametric forms.

Given that the coordinates of  are , find the values for  and  at the point of intersection.

Consider the two planes  and which can be defined by the equations 

Write down expressions for the normal vectors of each of the two planes.

Hence find the angle between the two planes. Give your answer in radians.

The points  and  have position vectors  and respectively, relative to the origin .

The position vectors are given by 

Find the direction vectors and .

Points ,  and  all lie on a single plane. 

Use the results from part (a) to write down the vector equation of the plane.

Find the Cartesian equation of the plane.

A plane lies parallel to the line with equation  and contains the points  and  with coordinates  and respectively. 

Find the vector .

By appropriate use of the vector product, find the normal to the plane.

Hence find the Cartesian equation of the plane.

Consider the plane defined by the Cartesian equation  

Show that the line with equation   lies in the plane.

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

Consider the planes  and , which are defined by the equations 

By solving the system of equations represented by the three planes show that the system of equations has a unique solution.

Hence write down the coordinates of any point(s) where all three planes intersect.

Consider the line  with vector equation and the plane  with Cartesian equation . 

Find the angle in radians between the line and the normal to the plane .

Hence find the angle in radians between the line  and the plane .

Two planes  and are defined by the equations 

Write down expressions for the normal vectors of each of the two planes.

Find the cross product of the two normal vectors.

Find the coordinates of a point that lies on both planes.

Hence find a vector equation of the line of intersection of the two planes.

A line  is defined by the Cartesian equation  and a plane  is defined by the Cartesian equation ,  where is a real constant. 

The line  lies in the plane .

Use the fact that the line  lies in the plane  to find the value of the constant .

Another line, , passes through the origin and is perpendicular to the plane .

Write down the equation of line  in vector form.

By considering the parametric form of the equation for , or otherwise, determine the point of intersection between line and the plane .

Hence determine the minimum distance between the plane  and the origin.

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.