Mathematics Revision Notes Of Three Dimensional Geometry For NDA
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Revision Notes Of Three Dimensional Geometry For NDA

Coordinates Axes and Coordinate Planes

The three mutually perpendicular lines in a space which divides the space into eight parts are called `"coordinate axes."` They determine three mutually perpendicular planes called `\color{orange} "coordinate planes."` The coordinates of a point arc the distances from the origin to the foot of the perpendiculars from the point on the respective coordinate axes. The coordinates of any point on X, Y and Z-axes will be as `(x,0,0) (0,y,0)` and `(0,0,z)` respectively and the coordinates of any point will be as `(x,y,z)`.

Distance between Two Points

The distance between two points `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)`

Section Formulae

There are two types of section formulae which are defined as follows:

For Internal Division

Formulae for internal division in cartesian and vector forms are given below:

`"(a) Cartesian Form :"`

Let two points `P (x_1 y_1, z_1 )` and `Q (x_2 , y_2 , z_2)` in space and `R(x,y,z)` be a point on the line joining points `P` and `Q` such that it divides the join of `P` and `Q` internally in the ratio `m: n`, then the coordinates of `R` are

`[(mx_2+ nx_1)/(m+n) , (my_2 +ny_1)/(m+n) , (mz_2 +nz_1)/(m+n)]`


`"(b) Vector Form : "`

Let `P` and `Q` be two points with position vectors `vecr_1` and `vecr_2` respectively and `OR= vecr`. Then

`vec r=(mvecr_2+nvecr_1)/(m+n)`

For External Division

Formulae for external division in Cartesian and vector forms are given below:

`"(a) Cartesian Form:"`

Let two points `P (x_1, y_1, z_1)` and `Q (x_2 , y_2 , z_2)` in space and `R` be a point on `PQ` produced dividing it externally in the ratio `m : n (m ne n )`. Then, the coordinates of `R` are

`[(mx_2-nx_1)/(m-n) , (my_2-ny_1)/(m-n) , (mz_2-nz_1)/(m-n)]`


(b) Vector Form :

Let `R` be a point dividing `PQ` externally in the ratio `m: n` and `OR= vecr`, then

`vecr= (m vecr_2-n vecr_1)/(m-n)`

Some Important Facts

If `A (x_1, y_1, z _1 ), B (x_2 , y_2 , z_2 )` and `C(x_3, y_3, z _3)` are the vertices of `Delta ABC`, then

(i) Centriod of Triangle `= ((x_1+x_2+x_3)/2 , (y_1+y_2+y_3)/3)`

(ii) Area of `Delta ABC=1/2 | ( hat i ,hatj,hat k), (x_2-x_1, y_2-y_1, z_2-z_1), (x_3-x_1, y_3-y_1, z_3-z_1)|`

(iii) If area of `Delta ABC = 0`, then these points are collinear.

Direction Cosines and Ratios



If a vector makes angles `alpha, beta` and `gamma` with the positive directions of `X, Y` and `Z`-axes respectively, then `cos alpha, cos beta` and `cos gamma` are called the direction cosines and it is denoted by `l, m, n` i.e. `l =cos alpha, m =cos beta` and `n =cos gamma`. If numbers `a, b` and `c` are proportional to `l, m` and `n` respectively, then `a, b` and `c` are called direction ratios.
Thus, `a, b` and `c` are the direction ratios of a vector, provided

`l/a =m/b =n/c`

where, `l= pm a/(sqrt (a^2+b^2 +c^2)), m = pm b/(sqrt(a^2+b^2+c^2))` and `n = pm c/(sqrt(a^2+b^2+c^2))`

Some Important Facts

(i) If `a` vector `vecr = a hat i + b hat j + c hat k` having direction cosines `l, m` and `n`,then

`l= a/(|r|) , m=b/(|r|)` and `n= c/(|r|)`

where, `a, b` and `c` are direction ratios of `r`.

(ii) Direction ratios of the line joining two points `P(x_1,y_1,z_1) ` and `Q(x_2 ,y_2 ,z_2 )` are `x_2 -x_1,y_2 -y_1,
z_2 - z_1` and its direction cosines are

`(x_2-x_1)/(|PQ|) , (y_2-y_1)/(|PQ|) , (z_2-z_1)/(|PQ|)`

(iii) lf `P (x, y,z)` is a point in space, then

(a) `x = l| r |, y = m | r|, z = n | r|`

(b) `l| r |, m | r |` and `n | r |` are projections of `r` on

`OX, OY` and `OZ`, respectively.

(c) `vec r = | r | (l hat i + m hat j + n hat k)`

(iv) The projection of the line segment joining points `P(x_1 , y_1, z_1)` and `Q(x_2 , y_2 , z_2)` to the line having
direction cosines `l,m, n,` is

`|(x_2 - x_1 )l + (y_2 - y_1 )m + (z_2 - z_1 )n|`.

(v) The sum of squares of direction cosines is always unity i.e. `l^2 + m^2 + n^2=1`

(vi) DC's of `X, Y` and `Z`-axes are `(1, 0, 0), (0, 1, 0)` and `(0, 0, 1)`.

(vii) Direction cosines of a line are unique but direction ratio of a line are not unique and can be infinite.

(viii) DC's of a line which is equally inclined to the coordinate axes, are `(pm 1/(sqrt 3) , pm 1/(sqrt 3) , pm 1/(sqrt 3))`

Straight Line in Space

A straight line is the locus of the intersection of any two planes.

Some Important Facts

Equation of a line passing through a given point and having given direction cosines

The equation of line which passes through `(x_1 ,y_1 ,z_1)` and having direction cosines `l, m, n,` is given by `(x-x_1)/l =(y-y_1)/m =(z-z_1)/n =r` (say)

If we use direction ratios instead of direction cosines, then the equation of the line is

`(x-x_1)/a =(y-y_1)/b =(z-z_1)/c= lambda`

where, `a^2 + b^2 + c^2 ne1`

This is also called the symmetrical form of a line.

Vector equation of a line passing through a point and parallel to given vector

The equation of a line passing through a point with position vector a and parallel to vector b, is `vecr = veca + lambda vecb`.

Equation of line passing through two given points

Cartesian Form : Direction ratios of `AB= (x_2-x_1 , y_2-y_1, z_2-z_1)`

`= (x_2- x_1, y_2- y_1, z_2- z_1)`

Direction ratios of `A P`

`= (x - x_1 , y -y_1 , z - z_1)`

Since, they are proportional.

`:. (x-x_1)/(x_2-x_1) =(y-y_1)/(y_2-y_1) =(z-z_1)/(z_2-z_1)`


Vector Form : `vecr =veca+ lambda (vecb- veca)`



Foot of perpendicular from a point ` (alpha, beta, gamma)` to the `(x-x_1)/l =(y-y_1)/m = (z-z_1)/n` length and equation of perpendicular

If `B` is the foot of perpendicular, then `B` is `(lk + x_1, mk + y_1 , nk + z_1 )`. Find the direction ratios of AB and apply the condition of
perpendicularity of `AB` and the given line. This will give the value of `k` and hence the point `B` which is
foot of perpendicular. Its length is the distance `AB` and its equation is the line joining two known points
`A` and `B`.

Angle between Two Intersecting Lines

The angle between two intersecting lines with cartesian equations and vector equations can be find as given below:

Cartesian Form

If `theta` is the angle between two lines whose direction cosines are `(l_1, m_1, n_1 ), (l_2,m_2,n_ 2 )` and `(a_1,b_1,c_1 ), (a_2,b_2 ,c_2)` are
direction ratios, then

`cos theta =l_1l_2 + m_1 m_2 + n_1 n_2`

`= pm (Sigma a_1a_2)/(|Sigma a_1^2 * Sigma a_2^2|^(1//2))`

`sin^2 theta = | (m_1 , n_1),(m_2 , n_2)|^2 + |(n_1 , l_1), (n_2 , l_2)|^2 +|(l_1, m_1), (l_2, m_2)|^2`

`tan theta = pm (| Sigma (l_1m_2-l_2 m_1)^2| ^(1//2))/(l_1l_2 +m_1m_2 +n_1n_2)`

The lines are parallel to each other if and only if

(i) `l_1/l_2 =m_1/m_2 =n_1/n_2`

(ii) `a_1/a_2 =b_1/b_2 =c_1/c_2`

The lines are perpendicular to each other if and only if

(i) `l_1l_2 + m_1m_2 + n_1n_2 = 0` or `theta = 90°`

(ii) `a_1a_2 + b_1b_2 + c_1 c_2 = 0`



Vector Form

If the vector equations of two lines

`vecr=veca_1 + lambda vecb_1 ` and `r= veca_2 + mu vecb_2` , then

`cos theta= | (vecb_1* vecb_2)/(|vecb_1| |vecb_2|)|`

Skew-lines

Two straight lines in a space which are neither parallel nor intersecting, are called skew-lines. Thus, skew-lines are those lines which do not lie in the same plane.

Shortest Distance between Two Lines :
Cartesian Form :
(i) Let the two skew-lines be `(x - x_1)/l_1 = (y-y_1)/m_1 = (z -z_1)/n_1` and `(x - x_2)/l_2 = (y - y_2)/m_2 = (z - z_2)/n_2`

`therefore d= ( | (x_2-x_1 , y_2-y_1 , z_2-z_1) , (l_1 , m_1 , n_1) , (l_2 , m_2 , n_2) |)/sqrt((m_1n_2-m_2n_1)^2+(n_1l_2-l_1n_2)^2+(l_1m_2-m_1l_2)^2)`


Vector Form : Shortest distance between the lines

`vecr=veca_1+lamda vecb_1` and `vecr = veca_2+muvecb_2` is given by `d=(|(veca_2-veca_1)*(vecb_1xxvecb_2)|)/(|(vecb_1xxvecb_2)|)`

Note : • The shortest distance between two parallel lines `vecr=veca_1+lamdavecb_1` and `vecr = veca_2+muvecb_2` is given by `d=(|(veca_2-veca_1)*(vecb_1xxvecb_2)|)/(|(vecb_1xxvecb_2)|)`
• Two lines `vecr_1 = veca_1 + lamda vecb_1` and `vecr_2 = veca_2 +mu vecb_2` will intersect provided `d = 0` i.e. when `(veca_2 -veca_1)·(vecb_1 xx vecb_2 ) = 0.`

Plane

A plane is a surface such that all the points of a straight line joining any two points on the surface lie on it.

Equation of Plane in Different Forms :

(i) General Equation of a Plane : Every equation of first degree in `x, y, z` represents a plane. Thus, the general equation of a plane is

`ax+by+cz+d = 0` where `a^2+b^2+c^2 ne 0`

Note : (a) Equation of `YZ`-plane is `x = 0`
(b) Equation of' `ZX`-plane is `y = 0`
(c) Equation of `XY`-plane is `z = 0`

(ii) Normal Form :

(a) Cartesian Form : If`l, m, n` are the direction cosines of the normal to a given plane and `p` is the length of
perpendicular from origin to the plane, then the equation of the plane is `lx+my+nz = p`

(b) Vector Form : If `hatn` is a unit vector normal to given plane and `p` is the length of perpendicular from the
origin to the plane, then the equation of the plane is `r * hatn = p`

(iii) One-point Form :

(a) Cartesian Form The equation of plane passing through one point `(x_1 , y_1, z_1 )` is `a(x- x_1 ) + b(y- y_r) + c(z- x_1) = 0`

(b) Vector Form If a is the position vector of the point through which plane passes, then the equation of plane is given by `(r - a)·hatn = 0.`

(iv) Intercepts Form of the Plane : The equation of a plane whose intercepts are `a, b` and `c` on `X, Y` and `Z`-axes, respectively is

`x/a+y/b+z/c = 1`

Equation of Planes in Particular Cases

I. The equation of plane parallel to the plane `ax + by + cz + d = 0` is `ax+ by+ cz + k = 0` [only constant term is changed]

2. Plane passes through two points `(x_1, y_1, z_1)` and `(x_2 , y_2 , z_2 )` is `a (x- x_1) + b (y - y_1) + c (z- z_1) = 0` where, `a (x_2 - x_1) + b (y_2 - y_1 ) + c (z_2 - z_1) = 0`

:3. Plane passes through three non-collinear points `(x_1 , y_1, z_1) , (x_2 , y_2, z_2)` and `(x_3, y_3, z_3)` is

`| (x-x_1 , y-y_1 , z-z_1) , (x_2-x_1 , y_2-y_1 , z_2-z_1) , ( x_3-x_1 , y_3-y_1 , z_3-z_1) | = 0`

4. Plane `ax +by+ cz + d = 0` intersecting a line segment joining `A (x_1, y_1 , z_1)` and `B (x_2 , y_2 , z_2)` divides in the ratio

`(ax_1+by_1+cz_1+d)/(ax_2+by_2+cz_2+d)`

(i) If this ratio is positive, then A and B are on opposite sides of the plane.
(ii) If this ratio is negative, then A and B are on the same side of the plane.

5. Any plane passing through the line of intersection of the planes `ax + by + cz + d = 0` and `a_1x + b_1y + c_1z + d_1 = 0` is `(ax +by+cz +d)+ lamda.(a_1x + b_1y + c_1z +d_1 )= 0`

6. Any plane containing the line `(x - x_1)/l = (y-y_1)/m = (z - z_1)/n` is `a(x-x_1) +b(y-y_1) +c(z-z_1) = 0` where `al+bm+cn = 0`

Some Important Facts :

(i) A line parallel to plane `ax+ by+ cz + d = 0` In this case, it will be perpendicular to normal. `therefore al+bm+cn = 0`

(ii) A line perpendicular to a plane `ax + by + cz + d = 0` In this case, it will be parallel to normal.

`therefore a/l = b/m = c/n `

(iii) Line to lie in the plane `ax +by+ cz + d = 0` The normal to the plane will be perpendicular to the line.

`therefore al+bm+cn = 0`

Also, point `(x_1, y_1 , z_1 )` through which the line passes will also lie on the plane.

`therefore ax_1+by_1+cz_1+d = 0`

Distance of a Point from a Plane

Distance of a point `(x_1 , y_1 , z_1)` from the plane `ax+by+cz +d = 0` is `(|ax_1+by_1+cz_1|)/sqrt(a^2+b^2+c^2)`

Distance of the origin is `(|d|)/sqrt(a^2+b^2+c^2)`

Distance between Two Parallel Planes

The distance between two parallel planes `ax+by+cz+d_1 = 0` and `ax+by+cz+d_2 = 0` is `(|d_1-d_2|)/sqrt(a^2+b^2+c^2)`

Coplanar Line

A line which is in the same plane as another line. Any two intersecting lines must lie in the same plane and therefore will be coplanar.

Condition for Coplanarity of Two Lines Cartesian Form :

The lines `(x-x_1)/l_1 = (y-y_1)/m_1 = (z -z_1)/n_1` and `(x-x_2)/l_2 = (y-y_2)/m_2 = (z-z_2)/n_2`

are coplanar, if `| (x_2-x_1 , y_2-y_1 , z_2-z_1) , (l_1 , m_1 , n_1) , (l_2 , m_2 , n_2) | = 0`

Vector Form :

Two lines `vecr = veca + lamda vecb` and `vecr = vecc + mu vecd` are coplanar or intersecting, if `(veca- vecc) · (vecb xx vecd)= 0`

`=> [veca qquad vecb qquad vecd ] = [ vecc qquad vecb qquad vecd]`

Angle between two Planes

Cartesian Form : The angle between the planes

`a_1x+b_1y+c_1z +d_1 = 0` and `a_2x+b_2y+c_2z+d_2 = 0` is given by

` costheta = pm (a_1 a_2 + b_1 b_2 + c_1 c_2)/sqrt((Sigmaa_1^2)sqrt(Sigma a_2^2))`

Vector Form : If `vecn_1` and `vecn_2` are normals to the plane and `theta` is the angle between `vecr · vecn_1 = d_1` and `vecr · vecn_2 = d_2`, then

`costheta = | (vecn_1 * vecn_2)/(| n_1 | |n_2|)|`

Angle between a Line and a Plane

Cartesian Form : If angle between the line `(x-x_1)/a = (y-y_1)/b = (z-z_1)/c` and the plane `a_1x+b_1y+c_1z +d = 0` is `theta` then `(90^0 - theta)` is the angle between normal and the line i.e. `cos(90^0 - theta) = | (aa_1 +b b_1+c c_1)/sqrt(a^2+b^2+c^2 sqrt(a_1^2+b_1^2+c_1^2))|`

Vector Form : The angle between a line `vecr = veca + lamda vecb` and plane `vecr · vecn = vecd` is defined as the complement of the angle between the
line and normal to the plane.

`sintheta = ( vecn * vecb)/(| n | | b|)`
 
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