Plane

Course Outlines

· Review the three dimensional Cartesian co-ordinates

· Cylindrical and spherical co-ordinates of a point

· General equation of first degree

· Linear equation of a plane

· Angle between two planes

· Angle between a line and a plane

· Plane through three points

· Plane through the intersection of two planes

· Length of perpendicular from a point to a plane

· Bisectors of angles between two planes

· Pair of planes

· Conditions for homogeneous second degree equation to represent a pair of planes

· Angle between two planes represented by a second degree homogeneous equation.

Coordinates in Space

In a plane the position of a point is determined by an ordered pair (x, y) of real numbers obtained with reference to two straight lines in the plane generally at right angles. The position of a point in space is, however, determined by an ordered triad (x, y, z) of real numbers.

Distance between Two Points

Direction Cosines of a Line

Let a, b, g be the angles which any line makes with the positive directions of the co-ordinates axes. Then cos a, cos b, cos g are called the direction cosines of the given line and are generally denoted by l, m, n respectively.

A useful relation If O be the origin of and (x, y, z) be the co-ordinates of a point P, then x = lr, y = mr, z = mr, l, m, n being the direction cosines of the line OP and r, the length of the segment OP.

Through the point P draw the line PL perpendicular to the X-axis so that OL = x.

From the right-angled triangle OPL, we have

Þ x/r = l Þ x = lr.

Similarly, we have

y = mr, z = nr

Relation Between Direction Cosines

If l, m, n are the direction cosines of a line,

l² + m² + n² = 1,

i.e., the sum of the squares of the direction cosines of every line is unity. Let OP be drawn through the origin parallel to the given line so that l, m, n are the cosines of the angles which the line OP makes with the co-ordinate axes OX, OY, OZ respectively.

Let (x, y, z) be the co-ordinates of any point P on this line.

Let OP = r.

We have x = lr, y = mr, z = nr.

Squaring and adding, we obtain

x² + y² + z² = (l² + m² + n²)r²

Þ r² = OP² = x² + y² + z² = (l² + m² + n²)r²

Þ l² + m² + n² = 1.

Conditions for perpendicularity and parallelism

(i) The given lines are perpendicular

Þ q = 90⁰

Þ cos q = 0

Þ a₁a₂ + b₁b₂ + c₁c₂ = 0.

(ii) The given lines are parallel

Þ the lines through the origin drawn parallel to the lines coincide,

Þ the direction cosines of the lines are the same.

Þ the direction ratios of the lines are proportional.

Worked Out Examples

Plane

A locus is said to be a plane if it is such that if P and Q are any two points on the locus, then every point on the line segment PQ is also a point on the locus.

General Equation of first Degree

An equation of the first degree x, y, z is of the form

ax + by + cz + d = 0,

where a, b, c and d are constants and a, b, c are not all zero. The condition that a, b, c are not all zero is equivalent to a² + b² + c² ¹ 0.

Theorem: Every equation of the first degree in x, y and z represents a plane.

Proof:

The general equation of first degree in x, y, z is

ax + by + cz + d = 0,

where a, b, c and d are constants and a, b, c are not all zero.

The locus of this equation will be a plane if every point of this line joining any two points on the locus also lies on the locus.

Let P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) be two points on the locus, so that

ax₁ + by₁ + cz₁ + d = 0 …..(i)