Analytic Geometry in 3D
Analytic Geometry in 3D
A. Points in 3D
Co-ordinates (x, y, z)
Section Formula; Distance between two points etc similar as 2D
Direction Cosines / Direction Ratio
Direction Cosines The angles a line segment made with the axes.
Example A1 (Theorem) Show that [pic].
Direction Ratio (the ratio of the direction cosines) [pic]
Corollary: If the direction ratio is [pic], then the direction cosines are
[pic], [pic] and [pic]
The direction ratio between two points [pic] and [pic] is:
[pic]
The angle between two lines is calculated using vector dot product.
Given two lines with direction ratio [pic]
Parallel [pic]
Perpendicular [pic]
B. Straight Lines in 3D
Symmetric Form
Let [pic] be a fixed point. If a variable point [pic] moves so that the direction ratio of the line PQ is constant (say [pic]). Then the locus of Q is a straight line in 3D.
Consider the direction ratio [pic]
Hence [pic]
This is called the symmetric form of a straight line on 3D (ref: point slope form in 2D)
Example B1 Find the equation of the line pass through [pic].
In general, the equation of the line joining [pic] and [pic] is
[pic] (ref: two point form in 2D)
Parametric Form
For a straight line [pic]
Then [pic] where t is any real number.
The parametric form is generally used in the calculation.
Example B2 Find the intersection pt. of [pic] and [pic].
Example B3 Show that [pic] and [pic]
are skew (non-parallel, non-intersecting) perpendicular.
Example B4 Find the foot of perpendicular from the point [pic] to the line [pic].
Example B5 Find the shortest distance between two skew lines.
[pic] and [pic]
C. Planes
Normal Form [pic]
where [pic] are the direction cosines of the normal of the plane,
and d is the perpendicular distance of the plane from the origin.
Example C1 3 points : [pic], [pic], [pic]
Example C2 1 point + 1 line: [pic], [pic]
Example C3 1 point + Normal: [pic], [pic]
Example C4 1 line([pic]) + parallel to another([pic]):
[pic] [pic]
Example C5 Foot of perpendicular from [pic] to [pic]
For a plane [pic]
Parallel planes [pic]
Perpendicular planes [pic]
Distance between a line and a plane [pic]
D. Miscellany
Example D1 Intersection between line and plane
[pic] and [pic].
Example D2 Line of intersection between two planes
[pic] and [pic].
Example D3 Verify that the point [pic] lies on the line [pic].
Example D4 Verify that the line [pic] lies on the plane [pic].
Example D5 Show that the lines [pic] and [pic]
are coplanar iff [pic]
Example D6 Area of Triangle
Find the area of the triangle ABC. [pic]
Example D7 Projection of a line on a plane.
Find the projection of [pic] on the plane [pic].
Exercise
A1 Find the centroid of the triangle [pic], [pic] and [pic].
B1 Show that [pic], [pic] and [pic] are collinear.
Hence find the equation of the line.
B2 Find the angle between the lines:
[pic] and [pic]
B3 [pic], [pic], [pic], [pic]
Find k if (a) [pic].
(b) [pic].
(c) BCD collinear.
C1 Find the line of intersection
[pic] and [pic].
C2 [pic]
[pic]
[pic]
Find the plane parallel to [pic] and pass through the intersection of [pic].
C3 Find the plane perpendicularly bisects the line segment from [pic] to [pic].
C4 Find the line pass through [pic] and parallel to both the planes
[pic] and [pic].
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