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Co-ordinate System, Plane, Straight Line and Solids of Revolution

December 8, 2016
Published By : Pratik Kataria
Categorised in:

Relations between three coordinate systems

Relations between cartesian and spherical polar system of coordinates are

x = r sinθ cosϕ

y = r sinθ sinϕ

z = r cosθ


Relations between cartesian and cylindrical system of coordinates are

x = ρ cosϕ

y = ρ sinϕ

z = z

Note:


(i) In cartesian coordinate system:

-∞ < x < ∞, -∞ < y < ∞, -∞ < z < ∞


(ii) In spherical polar system:

0 < r < ∞, 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π


(iii) In cylinderical coordinate system:

0 < ρ < ∞, 0 ≤ ϕ ≤ 2π, -∞ < z < ∞

Shift of origin

x’ = x – h


y’ = y – k


z’ = z – l

Distance formula

(a) Distance of a point P(x, y, z) from the origin

$OP = r = √{x^2 + y^2 + z^2}$

(b) Distance between two points

$P_1 Q_1 = √{(x_2 – x_1)^2 + (y_2 – y_1)^2 + (z_2 – z_1)^2}$

Division of the join of two given points

$x={mx_2 + nx_1}/{m+n}, y = {my_2 + ny_1}/{m+n}, z = {mz_2 + nz_1}/{m+n}$

Note: (i) if $m/n = λ$ or if R divides PQ in ratio λ : 1

$x = {λ x_2 + x_1}/{λ+1} , y = {λ y_2 + y_1}/{λ + 1} , z = {λ z_2 + z_1}/{λ + 1}$

(ii) If $m/n = λ$ is positive -> R divides line PQ internally

if negative – > R divides line PQ externally

Direction cosines

l = cosα , m = cosβ , n = cosγ

A useful result:

x = lr , y = mr , z = nr

Relation between direction cosines

$l^2 + m^2 + n^2 = 1$

$cos^2 α + cos^2 β + cos^2 γ = 1 $

Direction ratios

$l = a/{√{a^2 + b^2 + c^2}}$ , $m = b/{√{a^2 + b^2 + c^2}}$ , $n = c/{√{a^2 + b^2 + c^2}}$

Angle between two lines

$cos θ = l_1 l_2 + m_1 m_2 + n_1 n_2$

$cos θ = {a_1 a_2 + b_1 b_2 + c_1 c_2}/{√{a^2_1 + b^2_1 + c^2_1 } √{a^2_2 + b^2_2 + c^2_2}}$

Lagrange’s Identity: $(l^2_1 + m^2_1 + n^2_1) (l^2_2 + m^2_2 + n^2_2) – (l_1 l_2 + m_1 m_2 + n_1 n_2)^2 = (l_1 m_2 – l_2 m_1)^2 + (m_1 n_2 – m_2 n_1)^2 + (n_1 l_2 – n_2 l_1)^2$

Cor. 1: Expression for sinθ

$sinθ = √{(l_1 m_2 – l_2 m_1)^2 + (m_1 n_2 – m_2 n_1)^2 + (n_1 l_2 – n_2 l_1)^2}$

Cor. 2: Conditions for perpendicularity and parallelism



(a) When lines are perpendicular

$l_1 l_2 + m_1 m_2 + n_1 n_2 = 0$

$a_1 a_2 + b_1 b_2 + c_1 c_2 = 0$

(b) When lines are parallel

$l_1/l_2 = m_1/m_2 = n_1/n_2 = 1$

$a_1/a_2 = b_1/b_2 = c_1/c_2 $

Projection

(a) Projection of a point

(b) Projection of a segment of a line

(c) Length of the projection:


P’Q’ = PQ cosθ

Projection of join of two points on a line whose



Direction cosines are l, m, n

$P’Q’ = l (x_2 – x_1) + m(y_2 – y_1) + n(z_2 – z_1)$

The Plane

Equation of plane (Standard forms):

(a) General form: ax + by + cz + d = 0

Cor. 1: Equation of the plane passing through the point $(x_1, y_2, z_1)$

$a(x-x_1) + b(y-y_1) + c(z-z_1) = 0$

Cor. 2: General equation of a plane passing through origin is

ax + by + cz = 0

(b) Intercept form

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

(c)Normal (perpendicular) form

lx + my + nz = p

$p = d/{√{a^2 + b^2 + c^2}}$ gives length of perpendicular from origin


(d) Three points form:

$|{\table x,y,z,1;x_1,y_1,z_1,1;x_2,y_2,z_2,1;x_3,y_3,z_3,1}|$ = 0

(e) Equation of plane passing through common section of two planes

$a_1 x + b_1 y + c_1 z + d_1 + λ (a_2 x + b_2 y + c_2 z + d_2) = 0$

Angle between two planes

$cosθ = {a_1 a_2 + b_1 b_2 + c_1 c_2}/{√{a^2_1 + b^2_1 + c^2_1} √{a^2_2 + b^2_2 + c^2_2}}$

Cor.: Planes are perpendicular

$a_1 a_2 + b_1 b_2 + c_1 c_2 = 0$


Planes are parallel
$a_1/a_2 = b_1/b_2=c_1/c_2$

Length of the perpendicular

$p = {ax_1 + b y_1 + c z_1 + d}/{√{a^2 + b^2 + c^2}}$

The straight line

(a) General form

$a_1 x + b_1 y + c_1 z + d_1 = 0$

$a_2 x + b_2 y + c_2 z + d_2 = 0$

(b) Symmetrical form

${x-x_1}/l = {y-y_1}/m = {z-z_1}/n = r$


${x-x_1}/a = {y-y_1}/b = {z-z_1}/c = r$

Cor.: Coordinates of any point on the line are:

$(x_1 + lr, y_1 + mr, z_1 + nr)$ or $(x_1 + ar, y_1 + br, z_1 + cr)$

Line through two points

${x – x_1}/{x_2 – x_1} = {y – y_1}/{y_2 – y_1} = {z – z_1}/{z_2 – z_1}$

Coplanarity of lines

Condition: $|\table x_2 – x_1, y_2 – y_1, z_2 – z_1;l_1,m_1,n_1;l_2,m_2,n_2|$ = 0

For finding equation of plane, replace $x_2$ in above equation by x