1.4 3D Coordinate Geometry

Detailed Theory: Three-Dimensional Coordinate Geometry

1. The 3D Coordinate System

1.1 Introduction

The three-dimensional coordinate system extends the 2D Cartesian system by adding a third axis perpendicular to both x and y axes.

1.2 Components of the System

1. X-axis: Horizontal axis (usually pointing right)

2. Y-axis: Vertical axis (usually pointing up)

3. Z-axis: Axis perpendicular to both x and y axes (usually pointing out of the page/screen)

4. Origin (O): Point (0,0,0) where all three axes intersect

5. Coordinate Planes:

   • XY-plane: z = 0

   • YZ-plane: x = 0

   • ZX-plane: y = 0

6. Octants: Eight regions divided by the coordinate planes

   • Octant I: x>0, y>0, z>0

   • Octant II: x<0, y>0, z>0

   • Octant III: x<0, y<0, z>0

   • Octant IV: x>0, y<0, z>0

   • Octant V: x>0, y>0, z<0

   • Octant VI: x<0, y>0, z<0

   • Octant VII: x<0, y<0, z<0

   • Octant VIII: x>0, y<0, z<0

1.3 Coordinates of a Point

A point P in 3D space is represented by an ordered triple (x, y, z):

• x-coordinate: Distance from YZ-plane

• y-coordinate: Distance from ZX-plane

• z-coordinate: Distance from XY-plane

Example: Point A(2, 3, 4) means:

• 2 units from YZ-plane (positive = right of YZ-plane)

• 3 units from ZX-plane (positive = above ZX-plane)

• 4 units from XY-plane (positive = in front of XY-plane)

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2. Distance Formula in 3D

2.1 Distance Between Two Points

The distance between points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ is: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

Derivation: Extending Pythagoras theorem to 3D.

Example: Distance between A(1,2,3) and B(4,6,8): $d = \sqrt{(4-1)^2 + (6-2)^2 + (8-3)^2} = \sqrt{3^2 + 4^2 + 5^2} = \sqrt{9+16+25} = \sqrt{50} = 5\sqrt{2}$

2.2 Distance from Origin

Distance from origin to point P(x,y,z): $d = \sqrt{x^2 + y^2 + z^2}$

2.3 Applications

1. Collinearity in 3D: Three points A, B, C are collinear if: $AB + BC = AC$ or $AB + AC = BC$ or $AC + BC = AB$

2. Type of triangle in space: Similar to 2D but with 3D distances

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3. Section Formula in 3D

3.1 Internal Division

If point P divides line segment AB internally in ratio m:n, where A(x₁,y₁,z₁) and B(x₂,y₂,z₂), then: $P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right)$

Example: Point dividing A(1,2,3) and B(4,6,8) in ratio 2:3 internally: $P\left(\frac{2\times4 + 3\times1}{2+3}, \frac{2\times6 + 3\times2}{2+3}, \frac{2\times8 + 3\times3}{2+3}\right) = \left(\frac{8+3}{5}, \frac{12+6}{5}, \frac{16+9}{5}\right) = \left(\frac{11}{5}, \frac{18}{5}, 5\right)$

3.2 External Division

If point P divides AB externally in ratio m:n (m≠n), then: $P\left(\frac{mx_2 - nx_1}{m-n}, \frac{my_2 - ny_1}{m-n}, \frac{mz_2 - nz_1}{m-n}\right)$

3.3 Midpoint Formula

Midpoint of A(x₁,y₁,z₁) and B(x₂,y₂,z₂): $M\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)$

3.4 Centroid of Tetrahedron

For tetrahedron with vertices A(x₁,y₁,z₁), B(x₂,y₂,z₂), C(x₃,y₃,z₃), D(x₄,y₄,z₄): $G\left(\frac{x_1+x_2+x_3+x_4}{4}, \frac{y_1+y_2+y_3+y_4}{4}, \frac{z_1+z_2+z_3+z_4}{4}\right)$

3.5 Centroid of Triangle in 3D

For triangle with vertices A(x₁,y₁,z₁), B(x₂,y₂,z₂), C(x₃,y₃,z₃): $G\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{z_1+z_2+z_3}{3}\right)$

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4. Direction Cosines and Direction Ratios

4.1 Direction Cosines

For a line making angles α, β, γ with positive x, y, z axes respectively:

• $\cos\alpha = l$

• $\cos\beta = m$

• $\cos\gamma = n$

Properties:

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

2. Direction cosines of x-axis: (1,0,0)

3. Direction cosines of y-axis: (0,1,0)

4. Direction cosines of z-axis: (0,0,1)

4.2 Direction Ratios

Three numbers a, b, c proportional to direction cosines: $\frac{l}{a} = \frac{m}{b} = \frac{n}{c} = \frac{1}{\sqrt{a^2 + b^2 + c^2}}$

Relation between direction cosines and ratios: $l = \frac{a}{\sqrt{a^2 + b^2 + c^2}}, \quad m = \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \quad n = \frac{c}{\sqrt{a^2 + b^2 + c^2}}$

4.3 Direction Cosines of Line Through Two Points

For line through P(x₁,y₁,z₁) and Q(x₂,y₂,z₂), direction ratios are: $(x_2-x_1, y_2-y_1, z_2-z_1)$

Direction cosines are: $\left(\frac{x_2-x_1}{d}, \frac{y_2-y_1}{d}, \frac{z_2-z_1}{d}\right)$ where $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$

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5. Angle Between Two Lines

5.1 Angle Using Direction Cosines

If two lines have direction cosines (l₁,m₁,n₁) and (l₂,m₂,n₂), the angle θ between them is: $\cos\theta = l_1l_2 + m_1m_2 + n_1n_2$

Special Cases:

1. Parallel lines: $l_1 = l_2, m_1 = m_2, n_1 = n_2$

2. Perpendicular lines: $l_1l_2 + m_1m_2 + n_1n_2 = 0$

5.2 Angle Using Direction Ratios

If two lines have direction ratios (a₁,b₁,c₁) and (a₂,b₂,c₂), then: $\cos\theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2}\sqrt{a_2^2 + b_2^2 + c_2^2}}$

Example: Find angle between lines with direction ratios (1,2,3) and (4,5,6): $\cos\theta = \frac{1\times4 + 2\times5 + 3\times6}{\sqrt{1^2+2^2+3^2}\sqrt{4^2+5^2+6^2}} = \frac{4+10+18}{\sqrt{14}\sqrt{77}} = \frac{32}{\sqrt{1078}} \approx 0.9746$ $\theta \approx \cos^{-1}(0.9746) \approx 12.9^\circ$

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6. Projection of a Line Segment

6.1 Projection on Coordinate Axes

Projection of line segment PQ on:

• x-axis: $|x_2 - x_1|$

• y-axis: $|y_2 - y_1|$

• z-axis: $|z_2 - z_1|$

6.2 Projection on a Line with Given Direction Cosines

Projection of PQ with direction cosines (l,m,n): $\text{Projection} = |(x_2-x_1)l + (y_2-y_1)m + (z_2-z_1)n|$

Example: Projection of segment from (1,2,3) to (4,6,8) on line with direction cosines (1/√3, 1/√3, 1/√3): $\text{Proj} = \left|(4-1)\frac{1}{\sqrt{3}} + (6-2)\frac{1}{\sqrt{3}} + (8-3)\frac{1}{\sqrt{3}}\right| = \left|\frac{3+4+5}{\sqrt{3}}\right| = \frac{12}{\sqrt{3}} = 4\sqrt{3}$

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7. Equations of Lines in 3D

7.1 Vector Form

Line through point with position vector $\vec{a}$ and parallel to vector $\vec{b}$: $\vec{r} = \vec{a} + \lambda\vec{b}$ where λ is a scalar parameter.

7.2 Cartesian Form

a) Symmetric Form (Two-Point Form)

Line through (x₁,y₁,z₁) and (x₂,y₂,z₂): $\frac{x-x_1}{x_2-x_1} = \frac{y-y_1}{y_2-y_1} = \frac{z-z_1}{z_2-z_1}$

b) Symmetric Form (Direction Ratios)

Line through (x₁,y₁,z₁) with direction ratios a,b,c: $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$

c) Parametric Form

Line through (x₁,y₁,z₁) with direction ratios a,b,c:

$$\begin{cases} x = x_1 + aλ \\ y = y_1 + bλ \\ z = z_1 + cλ \end{cases}$$

where λ is parameter.

7.3 Conversion Between Forms

Example: Convert line $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ to parametric form: $x = 1 + 2λ, \quad y = 2 + 3λ, \quad z = 3 + 4λ$

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8. Angle Between Lines

8.1 Formula Using Direction Ratios

For lines: $L_1: \frac{x-x_1}{a_1} = \frac{y-y_1}{b_1} = \frac{z-z_1}{c_1}$ $L_2: \frac{x-x_2}{a_2} = \frac{y-y_2}{b_2} = \frac{z-z_2}{c_2}$

Angle θ between them: $\cos\theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$

8.2 Special Cases

1. Parallel lines: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

2. Perpendicular lines: $a_1a_2 + b_1b_2 + c_1c_2 = 0$

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9. Shortest Distance Between Lines

9.2 Distance Between Parallel Lines

Given two parallel lines:

Line 1: $\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$

Line 2: $\frac{x - x_2}{a} = \frac{y - y_2}{b} = \frac{z - z_2}{c}$

Vector Form Formula: The distance between these parallel lines is: $d = \frac{|(\vec{a}_2 - \vec{a}_1) \times \vec{b}|}{|\vec{b}|}$

Where:

• $\vec{a}_1 = (x_1, y_1, z_1)$ (position vector of point on line 1)

• $\vec{a}_2 = (x_2, y_2, z_2)$ (position vector of point on line 2)

• $\vec{b} = (a, b, c)$ (direction vector of both lines)

Cartesian Formula: The distance can also be calculated using: $d = \sqrt{\frac{D_1^2 + D_2^2 + D_3^2}{a^2 + b^2 + c^2}}$

Where: $D_1 = \begin{vmatrix} y_2 - y_1 & z_2 - z_1 \\ b & c \end{vmatrix}$ $D_2 = \begin{vmatrix} z_2 - z_1 & x_2 - x_1 \\ c & a \end{vmatrix}$ $D_3 = \begin{vmatrix} x_2 - x_1 & y_2 - y_1 \\ a & b \end{vmatrix}$

Expanded Form: $d = \frac{\sqrt{(b(z_2 - z_1) - c(y_2 - y_1))^2 + (c(x_2 - x_1) - a(z_2 - z_1))^2 + (a(y_2 - y_1) - b(x_2 - x_1))^2}}{\sqrt{a^2 + b^2 + c^2}}$

Example: Find the distance between the parallel lines:

Line 1: $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$

Line 2: $\frac{x - 4}{2} = \frac{y - 5}{3} = \frac{z - 6}{4}$

Solution:

Given:

• $x_1 = 1, y_1 = 2, z_1 = 3$

• $x_2 = 4, y_2 = 5, z_2 = 6$

• $a = 2, b = 3, c = 4$

Step 1: Calculate D₁, D₂, D₃

$D_1 = \begin{vmatrix} y_2 - y_1 & z_2 - z_1 \\ b & c \end{vmatrix} = \begin{vmatrix} 3 & 3 \\ 3 & 4 \end{vmatrix} = (3 \times 4) - (3 \times 3) = 12 - 9 = 3$

$D_2 = \begin{vmatrix} z_2 - z_1 & x_2 - x_1 \\ c & a \end{vmatrix} = \begin{vmatrix} 3 & 3 \\ 4 & 2 \end{vmatrix} = (3 \times 2) - (3 \times 4) = 6 - 12 = -6$

$D_3 = \begin{vmatrix} x_2 - x_1 & y_2 - y_1 \\ a & b \end{vmatrix} = \begin{vmatrix} 3 & 3 \\ 2 & 3 \end{vmatrix} = (3 \times 3) - (3 \times 2) = 9 - 6 = 3$

Step 2: Calculate the distance

$d = \sqrt{\frac{D_1^2 + D_2^2 + D_3^2}{a^2 + b^2 + c^2}} = \sqrt{\frac{3^2 + (-6)^2 + 3^2}{2^2 + 3^2 + 4^2}}$

$d = \sqrt{\frac{9 + 36 + 9}{4 + 9 + 16}} = \sqrt{\frac{54}{29}} = \frac{3\sqrt{6}}{\sqrt{29}}$

Final Answer: The distance between the parallel lines is $\frac{3\sqrt{6}}{\sqrt{29}}$ units.

9.3 Distance Between Skew Lines

For lines: $\vec{r} = \vec{a}_1 + \lambda\vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu\vec{b}_2$

Shortest distance $d = \frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)|}{|\vec{b}_1 \times \vec{b}_2|}$

Cartesian formula for lines: $L_1: \frac{x - x_1}{a_1} = \frac{y - y_1}{b_1} = \frac{z - z_1}{c_1}$ $L_2: \frac{x - x_2}{a_2} = \frac{y - y_2}{b_2} = \frac{z - z_2}{c_2}$

Distance $d = \frac{ \begin{vmatrix} x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} }{ \sqrt{ \begin{vmatrix} b_1 & c_1 \\ b_2 & c_2 \end{vmatrix}^2 + \begin{vmatrix} c_1 & a_1 \\ c_2 & a_2 \end{vmatrix}^2 + \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}^2 }}$

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10. Equations of Planes

10.1 General Equation

The general equation of a plane is: $Ax + By + Cz + D = 0$ where A, B, C are not all zero.

10.2 Normal Form

Plane with perpendicular distance p from origin and direction cosines of normal (l,m,n): $lx + my + nz = p$

10.3 Intercept Form

Plane making intercepts a, b, c on x, y, z axes: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$

10.4 Three-Point Form

Plane through points (x₁,y₁,z₁), (x₂,y₂,z₂), (x₃,y₃,z₃):

$$\begin{vmatrix} x - x_1 & y - y_1 & z - z_1 \\ x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\ x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \end{vmatrix} = 0$$

Alternative form:

$$\begin{vmatrix} x & y & z & 1 \\ x_1 & y_1 & z_1 & 1 \\ x_2 & y_2 & z_2 & 1 \\ x_3 & y_3 & z_3 & 1 \end{vmatrix} = 0$$

10.5 Equation from Point and Normal

Plane through (x₁,y₁,z₁) with normal having direction ratios (A,B,C): $A(x - x_1) + B(y - y_1) + C(z - z_1) = 0$

10.6 Equation from Intercepts

If plane cuts x-axis at (a,0,0), y-axis at (0,b,0), z-axis at (0,0,c): $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$

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11. Angle Between Planes

11.1 Using Normal Vectors

For planes: $A_1x + B_1y + C_1z + D_1 = 0$ and $A_2x + B_2y + C_2z + D_2 = 0$

Angle θ between them is angle between their normals: $\cos\theta = \frac{|A_1A_2 + B_1B_2 + C_1C_2|}{\sqrt{A_1^2 + B_1^2 + C_1^2}\sqrt{A_2^2 + B_2^2 + C_2^2}}$

11.2 Special Cases

1. Parallel planes: $\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} \neq \frac{D_1}{D_2}$

2. Perpendicular planes: $A_1A_2 + B_1B_2 + C_1C_2 = 0$

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12. Distance of a Point from a Plane

12.1 Distance Formula

Distance from point P(x₁,y₁,z₁) to plane $Ax + By + Cz + D = 0$: $d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}}$

Example: Distance from (1,2,3) to plane $2x - 3y + 6z - 5 = 0$: $d = \frac{|2(1) - 3(2) + 6(3) - 5|}{\sqrt{2^2 + (-3)^2 + 6^2}} = \frac{|2 - 6 + 18 - 5|}{\sqrt{4 + 9 + 36}} = \frac{9}{\sqrt{49}} = \frac{9}{7}$

12.2 Distance Between Parallel Planes

Distance between planes $Ax + By + Cz + D_1 = 0$ and $Ax + By + Cz + D_2 = 0$: $d = \frac{|D_1 - D_2|}{\sqrt{A^2 + B^2 + C^2}}$

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13. Angle Between Line and Plane

13.1 Formula

For line with direction ratios (a,b,c) and plane with normal ratios (A,B,C), if θ is angle between line and plane, and φ is angle between line and normal: $\sin\theta = \cos\phi = \frac{|Aa + Bb + Cc|}{\sqrt{A^2 + B^2 + C^2}\sqrt{a^2 + b^2 + c^2}}$

Alternative: If line makes angle α with plane, and direction cosines of line are (l,m,n) while normal to plane has direction cosines (L,M,N): $\sin\alpha = |lL + mM + nN|$

13.2 Special Cases

1. Line parallel to plane: $Aa + Bb + Cc = 0$

2. Line perpendicular to plane: $\frac{a}{A} = \frac{b}{B} = \frac{c}{C}$

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14. Sphere

14.1 Standard Equations

a) Center at Origin, Radius r

$x^2 + y^2 + z^2 = r^2$

b) Center at (h,k,l), Radius r

$(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2$

c) General Form

$x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0$ where center = $(-u, -v, -w)$ and radius = $\sqrt{u^2 + v^2 + w^2 - d}$

Conditions:

• Real sphere: $u^2 + v^2 + w^2 - d > 0$

• Point sphere: $u^2 + v^2 + w^2 - d = 0$

• Imaginary sphere: $u^2 + v^2 + w^2 - d < 0$

14.2 Equation with Diameter Ends

If (x₁,y₁,z₁) and (x₂,y₂,z₂) are endpoints of diameter: $(x - x_1)(x - x_2) + (y - y_1)(y - y_2) + (z - z_1)(z - z_2) = 0$

14.3 Plane Section of a Sphere

Intersection of sphere and plane is a circle.

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15. Cylinder

15.1 Right Circular Cylinder

Cylinder with axis as z-axis and radius r: $x^2 + y^2 = r^2$

15.2 General Cylinder

Cylinder with generators parallel to line with direction ratios (l,m,n) and guiding curve f(x,y)=0 in plane z=0: Equation obtained by eliminating λ from: $x = x_1 + l\lambda, \quad y = y_1 + m\lambda, \quad z = n\lambda$ where (x₁,y₁) satisfies f(x,y)=0.

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16. Cone

16.1 Right Circular Cone

Cone with vertex at origin, axis along z-axis, and semi-vertical angle α: $x^2 + y^2 = z^2\tan^2\alpha$

16.2 General Cone

Cone with vertex at (x₁,y₁,z₁) and guiding curve f(x,y)=0 in plane z=0: Equation obtained by eliminating λ from: $x = x_1 + l\lambda, \quad y = y_1 + m\lambda, \quad z = z_1 + n\lambda$ where (l,m,n) are direction ratios of generator.

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17. Solved Examples

Example 1: Distance and Section Formula

Find coordinates of point dividing line joining A(1,2,3) and B(4,6,8) in ratio 2:1 internally and externally.

Solution:

Internal division (m:n = 2:1): $P\left(\frac{2\times4 + 1\times1}{2+1}, \frac{2\times6 + 1\times2}{2+1}, \frac{2\times8 + 1\times3}{2+1}\right) = \left(\frac{8+1}{3}, \frac{12+2}{3}, \frac{16+3}{3}\right) = \left(3, \frac{14}{3}, \frac{19}{3}\right)$

External division (m:n = 2:1): $P\left(\frac{2\times4 - 1\times1}{2-1}, \frac{2\times6 - 1\times2}{2-1}, \frac{2\times8 - 1\times3}{2-1}\right) = \left(\frac{8-1}{1}, \frac{12-2}{1}, \frac{16-3}{1}\right) = (7, 10, 13)$

Example 2: Equation of Plane

Find equation of plane through points A(1,1,1), B(1,-1,1), C(-1,-1,1).

Solution: Using three-point form:

$$\begin{vmatrix} x - 1 & y - 1 & z - 1 \\ 1 - 1 & -1 - 1 & 1 - 1 \\ -1 - 1 & -1 - 1 & 1 - 1 \end{vmatrix} = 0$$

Simplify:

$$\begin{vmatrix} x - 1 & y - 1 & z - 1 \\ 0 & -2 & 0 \\ -2 & -2 & 0 \end{vmatrix} = 0$$

Expand determinant: $(x-1)[(-2)(0) - (0)(-2)] - (y-1)[(0)(0) - (0)(-2)] + (z-1)[(0)(-2) - (-2)(-2)] = 0$ $0 - 0 + (z-1)[0 - 4] = 0$ $-4(z-1) = 0$ $z = 1$

So the plane is $z = 1$ (horizontal plane).

Example 3: Angle Between Line and Plane

Find angle between line $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and plane $x + 2y + 3z = 4$.

Solution: Direction ratios of line: (2,3,4) Normal to plane: (1,2,3)

$\sin\theta = \frac{|2\times1 + 3\times2 + 4\times3|}{\sqrt{2^2+3^2+4^2}\sqrt{1^2+2^2+3^2}} = \frac{|2+6+12|}{\sqrt{4+9+16}\sqrt{1+4+9}} = \frac{20}{\sqrt{29}\sqrt{14}} = \frac{20}{\sqrt{406}}$

$\theta = \sin^{-1}\left(\frac{20}{\sqrt{406}}\right) \approx \sin^{-1}(0.992) \approx 82.8^\circ$

Example 4: Distance Between Skew Lines

Find shortest distance between lines: $L_1: \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ $L_2: \frac{x-2}{3} = \frac{y-4}{4} = \frac{z-5}{5}$

Solution: Given:

• For L₁: $x_1=1, y_1=2, z_1=3, a_1=2, b_1=3, c_1=4$

• For L₂: $x_2=2, y_2=4, z_2=5, a_2=3, b_2=4, c_2=5$

Step 1: Calculate the determinant in numerator:

$$\begin{vmatrix} x_2-x_1 & y_2-y_1 & z_2-z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} = \begin{vmatrix} 1 & 2 & 2 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{vmatrix}$$

Expanding: $= 1\begin{vmatrix}3 & 4 \\ 4 & 5\end{vmatrix} - 2\begin{vmatrix}2 & 4 \\ 3 & 5\end{vmatrix} + 2\begin{vmatrix}2 & 3 \\ 3 & 4\end{vmatrix}$ $= 1(15-16) - 2(10-12) + 2(8-9)$ $= 1(-1) - 2(-2) + 2(-1) = -1 + 4 - 2 = 1$

Step 2: Calculate the denominator:

First, calculate the three 2×2 determinants:

$\begin{vmatrix} b_1 & c_1 \\ b_2 & c_2 \end{vmatrix} = \begin{vmatrix} 3 & 4 \\ 4 & 5 \end{vmatrix} = 15 - 16 = -1$

$\begin{vmatrix} c_1 & a_1 \\ c_2 & a_2 \end{vmatrix} = \begin{vmatrix} 4 & 2 \\ 5 & 3 \end{vmatrix} = 12 - 10 = 2$

$\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} = \begin{vmatrix} 2 & 3 \\ 3 & 4 \end{vmatrix} = 8 - 9 = -1$

Denominator = $\sqrt{(-1)^2 + 2^2 + (-1)^2} = \sqrt{1 + 4 + 1} = \sqrt{6}$

Step 3: Calculate distance:

$d = \frac{|1|}{\sqrt{6}} = \frac{1}{\sqrt{6}}$

Final Answer: The shortest distance between the skew lines is $\frac{1}{\sqrt{6}}$ units.

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18. Practice Tips for Exams

1. 3D Distance: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$

2. Section Formula: Internal: $\left(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}, \frac{mz_2+nz_1}{m+n}\right)$

3. Direction Cosines: $l^2+m^2+n^2=1$

4. Angle between lines: $\cos\theta = l_1l_2+m_1m_2+n_1n_2$

5. Plane equation: General form: $Ax+By+Cz+D=0$

6. Distance point to plane: $d = \frac{|Ax_1+By_1+Cz_1+D|}{\sqrt{A^2+B^2+C^2}}$

7. Angle line-plane: $\sin\theta = \frac{|Aa+Bb+Cc|}{\sqrt{A^2+B^2+C^2}\sqrt{a^2+b^2+c^2}}$

8. Sphere: Center $(-u,-v,-w)$, radius $\sqrt{u^2+v^2+w^2-d}$

9. Visualization: Always try to visualize in 3D

10. Check octants: Consider signs of coordinates

11. Cross product: Useful for normals and distances

12. Determinants: Use for area/volume calculations

This comprehensive theory covers all aspects of 3D Coordinate Geometry with detailed explanations and examples, providing complete preparation for the entrance examination.