3D Geometry
The topic of three dimensional geometry is quite important and a bit complicated as compared to its counterpart two dimensional geometry. One of the ways of describing a 3D object is by approximating or assuming its shape as a mesh of triangles. A triangle is generally defined by three vertices wherein the positions of the vertices are described by the coordinates x, y and z.
The major heads that are included in 3D coordinate geometry are the direction ratios and direction cosines of a line segment. Various topics covered in this chapter include:
What do we mean by direction cosines of a line segment?
The direction cosines are the cosines of the angles between a line and the coordinate axis. If we have a vector (a, b, c) in three dimensional space, then the direction cosines of the vector are defined as
cos α = a/ √(a^{2} + b^{2} + c^{2})
cos β = b/ √(a^{2} + b^{2} + c^{2})
cos γ = c/ √(a^{2} + b^{2} + c^{2})
What are direction ratios?
If l, m and n are the direction cosines then the direction ratios say a, b and c are given by
Some key points:
- While the direction cosines of a line segment are always unique, the direction ratios are never unique and in fact they can be infinite in number.
- If the direction cosines of a line are l, m and n then they satisfy the relation l^{2} + m^{2} + n^{2} = 1.
- If the direction cosines of a line segment AB are (l, m, n) then those of line BA will be (-l, -m, -n).
Angle Between two lines
Let us assume that θ is the angle between the two lines say AB and AC whose direction cosines are l_{1}, m_{1} and n_{1} and l_{2}, m_{2} and n_{2} then
cos θ = l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2}
Also if the direction ratios of two lines a_{1}, b_{1} and c_{1} and a_{2}, b_{2} and c_{2} then the angle between two lines is given by
cos θ = (a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2})/ √( a_{1}^{2} + b_{1}^{2} + c_{1}^{2}) . √ (a_{2}^{2} + b_{2}^{2} + c_{2}^{2})
View the following video on direction cosines
What is the condition for parallel or perpendicular lines?
When the two lines are perpendicular, the angle between the lines is 90° which gives the condition of perpendicularity as
l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2} = 0
or this implies a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0.
Similarly, when the two lines are parallel, the angle between them i.e. θ = 0.
This gives l_{1}/l_{2} = m_{1}/m_{2} = n_{1}/n_{2}
This also gives a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2}
What is the projection of a line segment on a given line?
Suppose we have a line segment joining the points P (x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}), then the projection of this line on another line having direction cosines as l, m, n is AB = l(x_{2}-x_{1}) + m + m(y_{2}-y_{1}) + n(z_{2}-z_{1}).
What exactly is a sphere?
A sphere is basically a circle in three dimensions. Just as a circle, sphere is also perfectly round and is defined as the locus of points that lie at the same distance say ‘r’ from a fixed point in three dimensional space. This ‘r’ is called as the radius of the sphere and this fixed point is called the center of the sphere. The general equation of sphere in 3D is
(x-a)^{2} + (y-b)^{2} + (z-c)^{2} = r^{2}, where (a, b, c) is the center and r is the radius.
Illustration: The number of all possible triplets (a_{1}, a_{2}, a_{3}) such that a_{1} + a_{2} cos(2x) + a_{3} sin^{2}x = 0 for all x is zero/infinite/one/ three?
Solution: The given condition is a_{1} + a_{2} cos(2x) + a_{3} sin^{2}x = 0
If we put x = 0 and x = π/2 we get
a_{1} + a_{2} = 0 and a_{1} - a_{2} + a_{3 }= 0
Hence, a_{2} = - a_{1} and a_{3 }= a_{2} – a_{1} = -2a_{1}
And so this leads us to the equation of the form
a_{1} – a_{1} cos(2x) - 2a_{1} sin^{2}x = 0 for all x
or a_{1} (1 – cos(2x) - 2 sin^{2}x) = 0 for all x
which is satisfied for all values of a_{1}
Hence, infinite number of triplets (a_{1}-a_{1}-2a_{1}) are possible.
Illustration: Find the value of k such that (x-4)/1 = (y-2)/1 = (z-k)/2 lies in the plane 2x - 4y + z = 7.
Solution: Given equation of the straight line is
(x-4)/1 = (y-2)/1 = (z-k)/2
Since the line lies in the plane 2x - 4y + z = 7
Hence, the point (4, 2, k) must satisfy the plane which yields
8-8+k = 7
Hence, k = 7.
askIITians is a platform where students get the opportunity of asking any kind of doubts on topics like shortest distance between two lines, equation of plane parallel to xy plane or direction cosines of line along with the solutions of various 3D coordinate geometry problems.
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