G12 Chapter 3: Analytical Solid Geometry

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**(Useful Formulas)**

Distance Formula:
The distance $d$ between two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ is given by: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$
Two-Point Form:
Equation of a line passing through $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$: $$\frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1}$$

• General Equation: $Ax + By + Cz + D = 0$, where $(A, B, C)$ represent the direction ratios of the normal vector.
• One-Point Form: Passing through $(x_1, y_1, z_1)$: $$A(x - x_1) + B(y - y_1) + C(z - z_1) = 0$$
• Intercept Form: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$
• Normal Form: $lx + my + nz = p$ where $p$ is the distance from the origin.
• Perpendicular Distance: From $(x_0, y_0, z_0)$ to the plane: $$d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}}$$

Chapter 3: Analytical Solid Geometry
Summary & Formula Reference

Analytical solid geometry extends algebraic principles into three-dimensional space (3D space) using a right-handed Cartesian coordinate system defined by three mutually perpendicular axes: X, Y, and Z.

3.1 Coordinates of a Point in Space

• Coordinate Planes: The three axes intersect at the origin $(0, 0, 0)$ and form three coordinate planes: the XY-plane ($z=0$), YZ-plane ($x=0$), and ZX-plane ($y=0$). These planes divide space into eight octants.
• Point Representation: Any point $P$ in space is uniquely identified by an ordered triplet $(x, y, z)$.
• Distance Formula: The distance $d$ between two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ is given by: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$
• Section Formula: The coordinates of a point dividing the line segment joining $P_1$ and $P_2$ in the ratio $m:n$ internally are: $$\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right)$$
• Direction Cosines (Hyperbolic/Linear) & Ratios:
  • If a line makes angles $\alpha, \beta, \gamma$ with the positive X, Y, Z axes respectively, its direction cosines are $l = \cos\alpha$, $m = \cos\beta$, $n = \cos\gamma$, satisfying: $$l^2 + m^2 + n^2 = 1$$
  • Direction ratios $(a, b, c)$ are any numbers proportional to the direction cosines: $l = \frac{a}{\sqrt{a^2+b^2+c^2}}$, etc.

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3.2 Lines

A line in 3D space can be represented in various mathematical forms:

• Symmetrical Form (Cartesian Form): Equation of a line passing through $(x_1, y_1, z_1)$ with direction ratios $(a, b, c)$: $$\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} = r$$ Any general point on this line is given by $(x_1 + ar, y_1 + br, z_1 + cr)$.
• Two-Point Form: Equation of a line passing through $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$: $$\frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1}$$
• Vector Form: $\Vec{r} = \Vec{a} + \lambda\vec{b}$, where $\Vec{a}$ is the position vector of a point on the line, and $\Vec{b}$ is a vector parallel to the line.

3.3 Parallel, Skew, and Perpendicular Lines

For two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$:

• Parallel Lines: Proportional direction ratios: $$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$$
• Perpendicular Lines: Orthogonal orientation: $$a_1a_2 + b_1b_2 + c_1c_2 = 0$$
• Angle Between Two Lines: Given by: $$\cos\theta = \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2}\sqrt{a_2^2 + b_2^2 + c_2^2}}$$
• Skew Lines: Lines that are neither parallel nor intersecting. They lie in different planes.
  • Shortest Distance (SD): For $\Vec{r} = \vec{a}_1 + \lambda\vec{b}_1$ and $\Vec{r} = \vec{a}_2 + \mu\vec{b}_2$: $$SD = \frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)|}{|\vec{b}_1 \times \vec{b}_2|}$$
  • If $SD = 0$, the lines intersect and are coplanar.

3.4 Planes

• General Equation: $Ax + By + Cz + D = 0$, where $(A, B, C)$ represent the direction ratios of the normal vector.
• One-Point Form: Passing through $(x_1, y_1, z_1)$: $$A(x - x_1) + B(y - y_1) + C(z - z_1) = 0$$
• Intercept Form: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$
• Normal Form: $lx + my + nz = p$ where $p$ is the distance from the origin.
• Perpendicular Distance: From $(x_0, y_0, z_0)$ to the plane: $$d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}}$$

3.5 Spheres

• Central Form: $(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2$
• General Equation: $x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0$
  • Center: $(-u, -v, -w)$
  • Radius: $r = \sqrt{u^2 + v^2 + w^2 - d}$
• Diameter Form: $(x - x_1)(x - x_2) + (y - y_1)(y - y_2) + (z - z_1)(z - z_2) = 0$
• Intersection with a Plane: Forms a circle with radius $r_c = \sqrt{R^2 - p^2}$, where $R$ is the sphere's radius and $p$ is the perpendicular distance from the center to the plane.

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Example 1.

Find the equation of the line through the point $(-3, 5, 7)$ and perpendicular to

(a) $xy$-plane          (b) $yz$-plane          (c) $zx$-plane.

Find the point of intersection of the line and plane.

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Solution

(a) The equation of the line through the point $(-3, 5, 7)$ and perpendicular to $xy$-plane is $$x = -3, y = 5 .$$ The point of intersection of the line and $xy$-plane is $(-3, 5, 0)$.

(b) The equation of the line through the point $(-3, 5, 7)$ and perpendicular to $yz$-plane is $$y = 5, z = 7 .$$ The point of intersection of the line and $yz$-plane is $(0, 5, 7)$.

(c) The equation of the line through the point $(-3, 5, 7)$ and perpendicular to $zx$-plane is $$x = -3, z = 7 .$$ The point of intersection of the line and $zx$-plane is $(-3, 0, 7)$.

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Exercise 3.1

Question 1.

Find the equation of the plane containing the point (1,-2,3) and parallel to
(a) $xy$-plane

Short Answer$z=3$

(b) $yz$-plane

Short Answer$x=1$

(c) $ zx$- plane.

Short Answer$y=-2$

Question 2.

Find the equation of the line through the point (2,3,-4) and perpendicular to
(a) $xy$-plane

Short Answer$x=2,y=3;(2,3,0)$

(b) $yz$-plane

Short Answer$y=3,z=-4;(0,3,-4)$

(c) $zx$-plane.

Short Answer$x=2,z=-4;(2,0,-4)$

Find the point of intersection of the line and plane.

Question 3.

Find the distance between the points (2,-3,5) and (7,5,-2).

Short Answer$\sqrt{138}$

Question 4.

Show that the points (-1,2,5),(1,1,6) and (0,5,6) form a right triangle

Short Answer$AB^2 + CA^2 = 6 + 11 = 17 = BC^2$

Question 5.

Show that the points (1,-1,2),(3,-2,3) and (5,-3,4) are collinear.

Short Answer$\Vec{PQ} = \Vec{QR}$

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Example 2.

Given $P(1, 2, 3)$ and $Q(3, 6, 5)$, find the coordinates of point $R(x, y, z)$ on the line $PQ$ with respect to the point $P$ and the following parameters.

(a) $k = \frac{1}{2}$          (b) $k = 2$          (c) $k = -2$

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Solution

Given $P(1, 2, 3)$ and $Q(3, 6, 5)$, we have $\langle PQ \rangle = \langle l, m, n \rangle = \langle 2, 4, 2 \rangle$.

• (a) $k = \frac{1}{2} : \quad (x, y, z) = \left(1 + \frac{1}{2}(2), 2 + \frac{1}{2}(4), 3 + \frac{1}{2}(2)\right) = (2, 4, 4)$
• (b) $k = 2 : \quad (x, y, z) = (1 + 2(2), 2 + 2(4), 3 + 2(2)) = (5, 10, 7)$
• (c) $k = -2 : \quad (x, y, z) = (1 + (-2)(2), 2 + (-2)(4), 3 + (-2)(2)) = (-3, -6, -1)$

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Example 3.

Given $P(-1, 2, 3)$ and $Q(3, 5, -2)$, determine whether or not the following points are on the line $PQ$. If the point is on the line $PQ$, find the corresponding parameter with respect to the point $P$. \begin{align*} &\text{(a) } \left(1, \frac{7}{2}, \frac{1}{2}\right) &&\text{(b) } (7, 8, -7) &&\text{(c) } (-5, -1, 8) &&\text{(d) } (7, 8, -2) \end{align*}
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Example 4.

Given $P(2, 1, 3)$ and $Q(6, -5, 3)$, determine whether or not the following points are on the line $PQ$. If the point is on the line $PQ$, find the corresponding parameter with respect to the point $P$. \begin{align*} &\text{(a) } (4, -2, 3) &&\text{(b) } (-2, 7, 3) &&\text{(c) } (10, -11, 3) &&\text{(d) } (1, 1, 3) \end{align*}
Click to view Text Book Solution
Click to view detailed MM Solution ===============

Exercise 3.2

Question 1.

Given $P(3,1,5)$ and $Q(-3,7,-2)$, find the coordinates of point $R(x,y,z)$ on the line $PQ$ with respect to the point $P$ and the following parameters.
(a) $k=\frac{1}{2}\quad$

Short Answer$$R = \mathbf{\left(0, 4, \frac{3}{2}\right)}$$

(b) $k=3\quad$

Short Answer$$R = \mathbf{(-15, 19, -16)}$$

(c) $k=-2$

Short Answer$$R = \mathbf{(15, -11, 19)}$$

Question 2.

Given $P(-2,1,3)$ and $Q(4,4,-3)$, determine whether or not the following points are on the line $PQ.$ If the point is on the line $PQ$, fnd the corresponding parameter with respect to the point $P.$
(a) $6,3,-6)$

Short Answer$\textbf{not on the line}$

(b) $(6,5,-5)$

Short Answeron the line with parameter $\mathbf{k = \frac{4}{3}}$

(c) $(-4,0,5)$

Short Answeron the line with parameter $\mathbf{k = -\frac{1}{3}}$

(d) $(7,8,-2)$

Short Answer$\textbf{not on the line}$

Question 3.

Given $P(3,2,-1)$ and $Q(4,2,5)$, determine whether or not the following points are on the line $PQ.$ If the point is on the line $PQ$, find the corresponding real number with respect to the point $P.$
(a) $(5,2,11)$

Short Answeron the line with parameter $\mathbf{k = 2}$.

(b) $(2,2,-7)$

Short Answeron the line with parameter $\mathbf{k = -1}$

(c) $(\frac{7}{2},2,2)$

Short Answeron the line with parameter $\mathbf{k = \frac{1}{2}}$

(d) $(6,2,10)$

Short Answer$\textbf{not on the line}$

Question 4.

Find the points of intersection of the line joining the two points (2,4,5) and (3,5,-4)with the following planes.
(a) $xy$-plane

Short Answer$$\mathbf{\left(\frac{23}{9}, \frac{41}{9}, 0\right)}$$

(b) $yz$-plane

Short Answer$$\mathbf{(0, 2, 23)}$$

(c) $zx$-plane

Short Answer$$\mathbf{(-2, 0, 41)}$$

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