Three Dimensional Geometry

1. Distance formula

The distance between A$(x_1,y_1,z_1)$and B $(x_2,y_2,z_2)$ is

AB= $\sqrt{{(x_2-x_1)}^2,{(y_2-y_1)}^2+{(z_2-z_1)}^2}$

2. The coordinate of a point P(x, y, z) which divides the join of A($x_1$, $y_1$, $z_1$) and B ($x_2$, $y_2$, $z_2$)

a) Internally in the ration m:n is

x= $\frac{{\mathrm{mx}}_2+{\mathrm{nx}}_1}{m+n}$, y = $\frac{{\mathrm{my}}_2+{\mathrm{ny}}_1}{m+n}$, z = $\frac{{\mathrm{mz}}_2+{\mathrm{nz}}_1}{m+n}$

b) Externally in the ratio m:n is x = $\frac{{\mathrm{mx}}_2-{\mathrm{nx}}_1}{m-n}$, y = $\frac{{\mathrm{my}}_2-{\mathrm{ny}}_1}{m-n}$, z = $\frac{{\mathrm{mz}}_2-{\mathrm{nz}}_1}{m-n}$

3. The midpoint of the join of A ($x_1$, $y_1$, $z_1$) and B ($x_2$, $y_2$, $z_2$) is

x= $\frac{x_1+x_2}{2}$, y = $\frac{y_1+y_2}{2}$, z= $\frac{z_1+z_2}{2}$

4. Centroid of a triangle with vertices ($x_1$, $y_1$, $z_1$), ($x_2$, $y_2$, $z_2$), ($x_3$, $y_3$, $z_3$) is

$[\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3},\frac{z_1+z_2+z_3}{3}]$

5. Centroid of a tetrahedron with vertices ($x_1,y_1,z_1$) ($x_2,y_2,z_2$) ($x_3,y_3,z_3$) and ($x_4,y_4,z_4$) is $(\frac{\mathrm{\Sigma }x}{4},\frac{\mathrm{\Sigma }y}{4},\frac{\mathrm{\Sigma }z}{4})$

6. Direction cosines of a line making angles $\alpha$, $\beta$, $\gamma$ with the x, y, z axes respectively are l= cos $\alpha$, m = cos $\beta$, n = cos $\gamma$.

7. $l^2$+ $m^2$+$n^2$ = 1 = ${\cos }^2\alpha$ + ${\cos }^2\beta$ + ${\cos }^2\gamma$

8. D'Cs of the x, y, z axes are 1, 0, 0; 0, 1, 0; 0, 0, 1

9. If a, b, c are the Direction Ratios of a line, its D.C are $\frac{a}{\sqrt{{\mathrm{\Sigma }a}^2}}$, $\frac{b}{\sqrt{{\mathrm{\Sigma }a}^2}}$, $\frac{c}{\sqrt{{\mathrm{\Sigma }a}^2}}$

10. D.R.s of the line joining ($x_1,y_1,z_1$) and ($x_2,y_2,z_2$) are $x_2$- $x_1$, $y_2$- $y_1$, $z_2$- $z_1$

11. Projection of the line joining ($x_1,y_1,z_1$) and ($x_2,y_2,z_2$) on another line whose

D.C.s are l, m, n is ($x_2$- $x_1$) l + ${(y}_2$- $y_1$) m + ($z_2$- $z_1$) n

12. Angle between two lines whose D.C.s are $l_1$, $m_1$, $n_1$ and $l_2$, $m_2$, $n_2$ is given by

cos $\theta$= $l_1{l}_2$ + $m_1{m}_2$ + $n_1{n}_2$

${\sin }^2\theta$ = $\mathrm{\Sigma }{(l_1{m}_2-l_2{m}_1)}^2$


13. Condition for two lines to be perpendicular

i) $l_1{l}_2$
+ $m_1{m}_2$ + $n_1{n}_2$ = 0 If D.C.s are given

ii) $a_1{a}_2$
+ $b_1{b}_2$ + $c_1{c}_2$ = 0 If D.R.s are given

14. Condition for two lines to be parallel

i) $l_1{l}_2$
+ $m_1{m}_2$ + $n_1{n}_2$ = 0 If D.C.'s are given

ii) $\frac{a_1}{a_2}$= $\frac{b_1}{b_2}$=$\frac{c_1}{c_2}$if D.R's are given

15. General equation of a plane is ax + by + cz +d = 0

16. D.R s of the normal to the plane ax + by + cz + d= 0 are a, b, c

17. Equation of a plane passing through ($x_1,y_1,z_1$) is a( x - $x_1$) + b(y - $y_1$) + c(z- $z_1$)=

0.

18. Angle between the planes $a_{1}x$
+$b_{1}y$
+ $c_{1}z$+ $d_1$= 0 and $a_{2}x$
+$b_{2}y$
+ $c_{2}z$+ $d_2$= 0 is given by cos $\theta$= $\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{\mathrm{\Sigma }{a}_1^2}\sqrt{\mathrm{\Sigma }{a}_2^2}}$


19. Planes are perpendicular if $a_1{a}_2$+ $b_1{b}_2$+$c_1{c}_2$=0

Parallel if $\frac{a_1}{a_2}$=$\frac{b_1}{b_2}$=$\frac{c_1}{c_2}$

20. Intercept form: $\frac{x}{a}$+$\frac{y}{b}$+$\frac{z}{c}$=1

21. Normal form: lx + my + nz= p
22. Equation of a plane parallel to ax + by + cz + d= 0 is ax + by + cz + d= 0

23. Perpendicular distance of a point ($x_1,y_1,z_1$) from ax + by + cz+ d = 0 is $\frac{{\mathrm{ax}}_1+{\mathrm{by}}_1+{\mathrm{cz}}_1+d}{\sqrt{a^2+b^2+c^2}}$

24. Equation of planes bisecting the angle between two planes is $\frac{a_{1}x+b_{1}y+c_{1}z+d_1}{\sqrt{{a_1}^2+{b_1}^2+{c_1}^2}}$= $\pm \frac{a_{2}x+b_{2}y+c_{2}z+d_2}{\sqrt{{a_2}^2+{b_2}^2+{c_2}^2}}$

25. Equation of a plane passing through the line of intersection of the planes P= 0 and Q = 0 is P + $\lambda$Q = 0

26. Symmetrical form of a line is

$\frac{x-x_1}{l}$ = $\frac{y-y_1}{m}$=$\frac{z-z_1}{n}$ if l, m, n are the D.C.s

$\frac{x-x_1}{a}$= $\frac{y-y_1}{b}$ = $\frac{z-z_1}{c}$ if a, b, c are the D.R.s

27. Equation of a line passing through ($x_1,y_1,z_1$) and ($x_2,y_2,z_2$) is $\frac{x-x_1}{x_2-x_1}$= $\frac{y-y_1}{y_2-y_1}$= $\frac{z-z_1}{z_2-z_1}$

28. If $\frac{x-x_1}{l}$= $\frac{y-y_1}{m}$=$\frac{z-z_1}{n}$ is parallel to the plane ax + by + cz + d=0 then al + bm + cn = 0 and ${\mathrm{ax}}_1$ + ${\mathrm{by}}_1$+${\mathrm{cz}}_1$+d $\ne$ 0

29. If the line lies in the plane al + bm + cn = 0 and ${\mathrm{ax}}_1$ + ${\mathrm{by}}_1$+${\mathrm{cz}}_1$+d $=$ 0

30. If the line is perpendicular to the plane $\frac{l}{a}$= $\frac{m}{b}$=$\frac{n}{c}$

31. Equation of a plane containing the line $\frac{x-x_1}{l}$ = $\frac{y-y_1}{m}$=$\frac{z-z_1}{n}$ is a(x - $x_1$) + b(y - $y_1$) + c (z - $z_1$) = 0 where al + bm + cn= 0

32. Angle between a line and the plane is given by sin $\theta$= $\frac{\mathrm{al}+\mathrm{bm}+\mathrm{cn}}{\sqrt{\mathrm{\Sigma }{a}^2}\sqrt{\mathrm{\Sigma }{l}^2}}$

33. Equation of a plane through the line $\frac{x-x_1}{l_1}$ = $\frac{y-y_1}{m_1}$=$\frac{z-z_1}{n_1}$ and parallel to another line whose D.R.s are $l_2$, $m_2$, $n_2$ is $\left[\begin{array}{ccc}x-x_1& y-y_1& z-z_1\\ l_1& m_1& n_1\\ l_2& m_2& n_2\end{array}\right]$= 0

34. Method to find the image of a point in a given plane.

Step 1. Write the equation of the line passing through P and normal to the given plane

as $\frac{x-x_1}{a}$ = $\frac{y-y_1}{b}$=$\frac{z-z_1}{c}$.

Step 2. Write the coordinate of image Q as ($x_1$, lr, $y_1$ + mr, $z_1$ + nr)

Step 3. Find the mid-point r of PQ

Step 4. Use the condition that r lies on the given plane and find r.

Step 5. Putting the value of r in the co-ordinates of Q, we get the required image of P.

35. Any plane through the above line is

A(x - $x_1$) + B(y - $y_1$) + C(z - $z_1$) = 0 where Al + Bm + Cn= 0.

36. S.D between two skew lines is ($x_2$-$x_1$)l + ($y_2$-$y_1$)m + ($z_2$-$z_1$)n

37. Length of the perpendicular from a point P($\overline{r_1}$) upon the line $\overline{r}$= $\overline{a}$ + $\lambda$ $\overline{b}$= $\frac{|(\overline{a}-\overline{r_1})\times \overline{b}|}{\left|\overline{b}\right|}$

38. Shortest distance between lines $\overline{r}$= $\overline{a_1}$ + $\lambda$ $\overline{b}$ and $\overline{r}$ = $\overline{a_2}$ + $\mu$ $\overline{b}$ is = $\frac{|(\overline{a_2}-\overline{a_1})\times \overline{b}|}{\left|\overline{b}\right|}$

39. Shortest distance between lines $\overline{r}$= $\overline{a_1}$ + $\lambda$$\overline{b_1}$ and $\overline{r}$ = $\overline{a_2}$ + $\mu$$\overline{b_2}$ is d= $\frac{\left|(\overline{a_2}-\overline{a_1})\mathrm{.}(\overline{b_1}-\overline{b_2})\right|}{|\overline{b_1}\times \overline{b_2}|}$

40. Two coplanar lines $\overline{r}$= $\overline{a_1}$ + $\lambda$$\overline{b}$ and $\overline{r}$= $\overline{a_2}$ + $\mu$$\overline{b_2}$ will intersect if d = 0

i.e., if $(\overline{a_2}-\overline{a_1})$. $(\overline{b_1}\times \overline{b_2})$= 0

41. Bisectors of the angles between two st. lines $\overline{r}$= $\overline{a}$ + $\lambda$$\overline{b}$ and $\overline{r}$= $\overline{a}$ + $\mu$$\overline{c}$ are $\overline{r}$= $\overline{a}$ + t ($\overline{c}+\overline{b}$) and $\overline{r}$= $\overline{a}$ + t ($\overline{c}$ - $\overline{b}$)

42. $x^2$+$y^2$+$z^2$+ 2ux + 2vy + 2wx + d= 0 represents a sphere.

Center (-u, -v, -x) radius $\sqrt{u^2+v^2+w^2-d}$

43. Equation of the sphere on the line joining ($x_1$, $y_1,z_1$) and ($x_2$, $y_2,z_2$) $$ as diameter is : (x - $x_1$) (x - $x_2$) + (y - $y_1$) (y - $y_2$) + (z - $z_1$) (z - $z_2$)= 0

44. $x^2$+$y^2$+$z^2$+ 2ux + 2vy + 2wz + d= 0, lx + my + nz= p take together represents a circle.

45. Equation of a sphere passing through the circle

$x^2$+$y^2$+$z^2$+ 2ux + 2vy + 2wz + d= 0, lx + my + nz= p

$x^2$+$y^2$+$z^2$+ 2ux + 2vy + 2wz + d + $\lambda$ (lx + my + nz- p)= 0

46. Intersection of two spheres is a circle.

47. Equation of the tangent plane at ($x_1$, $y_1,z_1$) is

${\mathrm{xx}}_1$ + ${\mathrm{yy}}_1$ + ${\mathrm{zz}}_1$ + u( x+ $x_1$) + v (y+$y_1$) + w(z+$z_1$) + d = 0

48. Length of the tangent from ($x_1$, $y_1$, $z_1$) to sphere S= 0 is $\sqrt{S_1}$

49. Equation of the radical plane is $S_1$ - $S_2$ = 0.