Planes

The Equation of a plane

Suppose is a specific point in a plane and is a vector orthogonal to that plane. Now let be any other point in the plane.

Then the vector

must be orthogonal to

. So

which gives us an equation for the plane

For example, the equation

describes a plane in space that has a vector

orthogonal to it.

Red Question: Plane

passes through the point and has normal vector . Plane passes through the

points

and has normal vector . Find the intersection of these planes.

Green Question: Find the equation of the plane that contains the point

and the the vectors and

Blue Question: Find a unit vector normal to the plane whose intercepts are

, and .

Skill Set

Planes

No. 1

STEP 2000, Math II, #7: The line

has vector equation , where

and

is a scalar parameter. Find an expression for the angle between and the line . Show that there is a

line

through the origin such that, whatever the value of , the acute angle between and is .

A plane has equation . The line meets this plane at P. Show that, as varies, P describes a circle, with its centre on .

Find the radius of this circle.

STEP 2001, Math III, #6: The plane

meets the co-ordinate axes at the points A, B, and C. The point M has coordinates

and O is the origin.

Show that OM meets the plane at the centroid

of triangle ABC. Show also that the perpendiculars to the plane from O and from M meet the plane at the orthocentre and at the

circumcentre of triangle ABC respectively.

Hence prove that the centroid of a triangle lies on the line segment joining its orthocentre and circumcentre, and that it divides this

line segment in the ratio 2:1.

[The orthocentre of a triangle is the point at which the three altitudes intersect; the circumcentre of a triangle is the point equidistant

from the three vertices.]

STEP 1998, Math II, #8: Points A, B, and C in three-dimensions have coordinate vectors

, , , respectively. Show that the lines

joining the vertices of the triangle ABC to the mid-points of the opposite sides meet at a point R.

P is a point which is not in the plane ABC. Lines are drawn through the mid-points of BC, CA, and AB parallel to PA, PB, and PC respectively.

Write down the vector equation of the lines and show by inspection that these lines meet at a common point Q.

Prove further that the line PQ meets the plane ABC at R.

STEP 2002, Math II, #6: The lines , and lie in an inclined plane P and pass through a common point A. The line is a line of

greatest slope in P. The line

is perpendicular to and makes an acute angle with . The angles between the horizontal and ,

and

are , and , respectively.

Show that

and find the value of

. Deduce that .

The lines and are rotated in P about A so that and remain perpendicular to each other. The new acute angle between and

is . The new angles which and make with the horizontal are and , respectively. Show that

STEP 2000, Math I, #5: Arthur and Bertha stand at a point O on an inclined plane. The steepest line in the plane through O makes an angle

with the horizontal. Arthur walks uphill at a steady pace in a straight line which makes an angle with the steepest line. Bertha walks

uphill at the same speed in a straight line which makes an angle

with the steepest line (and is on the same side of the steepest line as

Arthur). Show that, when Arthur has walked a distance

, the distance between Arthur and Bertha is

Show also that, if , the line joining Arthur and Bertha makes an angle with the vertical, where

Page updated

Google Sites

Report abuse