G12 Chapter 3: Analytical Solid Geometry Summary

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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 $$x = -3, y = 5 \quad \text{or} \quad (-3, 5, z).$$ 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 $$y = 5, z = 7 \quad \text{or} \quad (x, 5, 7).$$ 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 $$x = -3, z = 7 \quad \text{or} \quad (-3, y, 7).$$ The point of intersection of the line and $zx$-plane is $(-3, 0, 7)$.

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*}
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Example 5.

Given $P(2,1,3)$, $Q(6,-5,4)$, $R(2,3,4)$ and $S(-1,5,4)$, determine whether the lines $PQ$ and $RS$ are parallel or skew or intersect.
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Example 6.

Given $P(0, 0, 1)$, $Q(3, 6, 4)$, $R(0, 3, 1)$ and $S(3, 0, 4)$, show that the lines $PQ$ and $RS$ are perpendicular.
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Example 7.

Find the equation of the line passing through the point $(-4, 7, -3)$ and perpendicular to the line \[ (x, y, z) = (3 + 2k, -1 + 3k, 1 - k). \] Find also the point of intersection of two lines.
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Example 8.

Find the equation of the plane containing $A(1,0,1)$, $B(3,6,4)$ and $C(-2,3,1)$.
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Example 9.

Find the equation of the line that passes through the point $(-1, 3, 2)$ and perpendicular to the plane $3x - 2y - z = 3$. Find the point of intersection of the line and the given plane.
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Example 10.

Find the equation of the plane containing the point $(-1, 3, 2)$ and parallel to the plane $3x - 2y - 3z = 2$.
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Example 11.

Find the equation of the plane tangent to the sphere \[ (x - 2)^2 + (y - 1)^2 + (z + 1)^2 = 14 \] at the point $(3, 4, 1)$.
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Example 12.

Find the equation of the sphere with center $(0, 1, 0)$ and touching the plane $x - 2y + 2z + 5 = 0$.
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Example 13.

Find the equation of a sphere that passes through the points $(9, 0, 0)$, $(3, 13, 5)$ and $(11, 0, 10)$, given that its center lies on the $yz$-plane.
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