(co-ordinates, lines, planes and spheres)
If a,b,c are proportional to l,m,n then a,b,c are called direction ratios of the line and
can be used as direction cosines.
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note:
If l,m,n are the direction cosines of a line,
then l 2+m 2+n 2 = 1
The direction ratios of a line passing through (x1,y1,z1) and (x2,y2,z2)
is given by (x2-x1) : ( y2-y1) : (z2-z1)
The projection of the line segment joining (x1,y1,z1) and (x2,y2,z2) on a line with direction cosines l ,m , n is given by l (x2-x1) + m( y2-y1) + n (z2-z1)
Straight line
represents a line passing through (x1,y1,z1) and having direction ratios a:b:c .
Any point on this line will be of the form.
Plane
ax +by + cz +d = 0 represents a plane whose normal has direction ratios a:b:c.
a (x-x1) + b (y-y1) +c (z-z1) = 0 represents a plane passing through (x1,y1,z1) and whose normal has direction ratios a:b:c.
perpendicular distance of (x1,y1,z1) from ax +by + cz +d = 0 is given by
Sphere
represents a sphere with centre at (x1,y1,z1) and radius r.
represents a sphere with centre at (-u,-v,-w) and whose radius is given by
.(x-x1)(x-x2) +(y-y1)(y-y2) + (z-z1)(z-z2) = 0 is the equation for which the line segement joining (x1,y1,z1) and (x2,y2,z2) is a diameter.