JEE Mathematics Three Dimensional Geometry Notes Set 05

Direction Cosines and Ratios

If a line makes angles \(\alpha,\beta,\gamma\) with the axes, its direction cosines are \(l=\cos\alpha,\ m=\cos\beta,\ n=\cos\gamma\) with \(l^2+m^2+n^2=1\), so \(\sin^2\alpha+\sin^2\beta+\sin^2\gamma=2\) and \(\cos2\alpha+\cos2\beta+\cos2\gamma=-1\). Direction ratios \(a,b,c\) give \(l,m,n=\frac{\pm(a,b,c)}{\sqrt{a^2+b^2+c^2}}\). The line through \((x_1,y_1,z_1)\) and \((x_2,y_2,z_2)\) has ratios \(x_2-x_1,\,y_2-y_1,\,z_2-z_1\). The angle between lines with ratios \((a_1,b_1,c_1)\), \((a_2,b_2,c_2)\): \(\cos\theta=\frac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{\sum a_1^2}\sqrt{\sum a_2^2}}\). A line equally inclined to the axes has \(l=m=n=\pm\frac1{\sqrt3}\).

Equation of a Plane

General form \(ax+by+cz+d=0\), where \((a,b,c)\) is the normal. Normal form \(lx+my+nz=p\) with p the perpendicular distance from the origin. Intercept form \(\frac xa+\frac yb+\frac zc=1\); the centroid of the triangle cut off on the axes is \(\left(\frac a3,\frac b3,\frac c3\right)\). Plane through three points: use the cross product of two edges as normal. Through the intersection of two planes \(P_1=0\), \(P_2=0\): \(P_1+\lambda P_2=0\). Vector form \(\mathbf r\cdot\mathbf n=d\).

Distances and Angles

Distance of \((x_1,y_1,z_1)\) from \(ax+by+cz+d=0\): \(\frac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}}\). Distance between parallel planes \(ax+by+cz+d_1=0\) and \(ax+by+cz+d_2=0\): \(\frac{|d_1-d_2|}{\sqrt{a^2+b^2+c^2}}\). Foot and image: the foot of the perpendicular from P is \(P-\frac{(\text{value})}{|\mathbf n|^2}\mathbf n\), and the image is \(P-\frac{2(\text{value})}{|\mathbf n|^2}\mathbf n\). Angle between two planes is the angle between their normals; the planes are perpendicular if \(\mathbf n_1\cdot\mathbf n_2=0\) and parallel if the normals are proportional.

Line and Plane Relations

A line with direction \((l,m,n)\) is parallel to the plane \(ax+by+cz+d=0\) if \(al+bm+cn=0\), and perpendicular if the ratios are proportional to \((a,b,c)\). Four points are coplanar when the scalar triple product of the three edge vectors from one of them is zero. The line of intersection of two planes has direction \(\mathbf n_1\times\mathbf n_2\).

Worked Examples

Example 1. Plane through \((2,2,1)\) and the intersection of \(3x-y+2z-4=0\) and \(x+y+z-2=0\): \(\lambda=-\frac23\) gives \(7x-5y+4z-8=0\).

Example 2. Distance between \(2x+3y+4z=4\) and \(4x+6y+8z=12\): rewrite the second as \(2x+3y+4z=6\), distance \(\frac{2}{\sqrt{29}}\).

Example 3. Reflection of \((2,-1,3)\) in \(3x-2y-z=9\) is \(\left(\frac{26}7,-\frac{15}7,\frac{17}7\right)\).

Quick Tips

Write every plane with a normal and a constant before comparing. Use the family \(P_1+\lambda P_2=0\) for conditions on a plane through a line.