Three Dimensional Geometry

The One Thing

Three-dimensional geometry becomes systematic when lines are represented by position and direction vectors. Direction cosines and dot products describe orientation, while cross products and scalar triple products determine perpendicularity, coplanarity, and shortest distances.

Definitions and Results

• Point in 3D: A point is represented by an ordered triple $(x,y,z)$, giving its coordinates along the three mutually perpendicular $x$-, $y$-, and $z$-axes.

• Position Vector: The position vector of $(x,y,z)$ from the origin is

$$x\mathbf{i}+y\mathbf{j}+z\mathbf{k},$$ where $\mathbf{i},\mathbf{j},\mathbf{k}$ are unit vectors along the coordinate axes.

• Direction Ratios: Three numbers $a,b,c$ proportional to the changes in the $x$-, $y$-, and $z$-coordinates along a line are called its direction ratios.

• Direction Cosines: If a line makes angles $\alpha ,\beta ,\gamma$ with the positive $x$-, $y$-, and $z$-axes respectively, then

$$l=\cos \alpha ,m=\cos \beta ,n=\cos \gamma$$ are its direction cosines.

• Relation Between Direction Cosines: The direction cosines satisfy

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

• Direction Cosines from Direction Ratios: If $a,b,c$ are direction ratios, then

$$l=\frac{a}{\sqrt{a^2+b^2+c^2}},m=\frac{b}{\sqrt{a^2+b^2+c^2}},n=\frac{c}{\sqrt{a^2+b^2+c^2}}.$$

• Equation of a Line in Vector Form: A line passing through a point with position vector $\mathbf{a}$ and having direction vector $\mathbf{b}$ is

$$\mathbf{r}=\mathbf{a}+\lambda \mathbf{b},$$ where $\lambda$ is a real parameter.

• Equation of a Line in Cartesian Form: A line through $(x_1,y_1,z_1)$ with direction ratios $a,b,c$ is

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

• Parametric Form of a Line: The same line may be written as

$$x=x_1+\lambda a,y=y_1+\lambda b,z=z_1+\lambda c.$$

• Line Through Two Points: The line joining $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ has direction ratios

$$x_2-x_1,y_2-y_1,z_2-z_1.$$ If $\mathbf{a}$ and $\mathbf{b}$ are the position vectors of the two points, its vector equation is $$\mathbf{r}=\mathbf{a}+\lambda (\mathbf{b}-\mathbf{a}).$$

• Distance Between Two Points: The distance between $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ is

$$\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}.$$

• Angle Between Two Lines: If the direction ratios are $(a_1,b_1,c_1)$ and $(a_2,b_2,c_2)$, the acute angle $\theta$ satisfies

$$\cos \theta =\frac{\left|a_1{a}_2+b_1{b}_2+c_1{c}_2\right|}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}.$$ Equivalently, for direction vectors ${\mathbf{b}}_1,{\mathbf{b}}_2$, $$\cos \theta =\frac{|{\mathbf{b}}_1\cdot {\mathbf{b}}_2|}{|{\mathbf{b}}_1||{\mathbf{b}}_2|}.$$

• Parallel Lines: Two lines are parallel when their direction vectors are scalar multiples of each other. Equivalently,

$${\mathbf{b}}_1\times {\mathbf{b}}_2=0,$$ or their direction ratios are proportional.

• Perpendicular Lines: Two lines are perpendicular when

$${\mathbf{b}}_1\cdot {\mathbf{b}}_2=0.$$

• Coplanar Lines: For

$$\mathbf{r}={\mathbf{a}}_1+\lambda {\mathbf{b}}_1,\mathbf{r}={\mathbf{a}}_2+\mu {\mathbf{b}}_2,$$ the lines are coplanar if $$({\mathbf{a}}_2-{\mathbf{a}}_1)\cdot ({\mathbf{b}}_1\times {\mathbf{b}}_2)=0.$$

• Skew Lines: Two lines are skew if they are neither parallel nor intersecting and do not lie in the same plane.

• Shortest Distance Between Intersecting Lines: If two lines intersect, their shortest distance is

$$d=0.$$

• Shortest Distance Between Parallel Lines: For

$$\mathbf{r}={\mathbf{a}}_1+\lambda \mathbf{b},\mathbf{r}={\mathbf{a}}_2+\mu \mathbf{b},$$ the shortest distance is $$d=\frac{|({\mathbf{a}}_2-{\mathbf{a}}_1)\times \mathbf{b}|}{|\mathbf{b}|}.$$

• Shortest Distance Between Skew Lines: For

$$\mathbf{r}={\mathbf{a}}_1+\lambda {\mathbf{b}}_1,\mathbf{r}={\mathbf{a}}_2+\mu {\mathbf{b}}_2,$$ the shortest distance is $$d=\frac{|({\mathbf{a}}_2-{\mathbf{a}}_1)\cdot ({\mathbf{b}}_1\times {\mathbf{b}}_2)|}{|{\mathbf{b}}_1\times {\mathbf{b}}_2|}.$$

• Common Perpendicular: The shortest segment joining two non-intersecting lines is perpendicular to both lines.

• Cross Product: The vector ${\mathbf{b}}_1\times {\mathbf{b}}_2$ is perpendicular to both ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$. For two non-parallel lines, it gives the direction of their common perpendicular.

• Scalar Triple Product: The expression

$$\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})$$ gives the volume of the parallelepiped formed by the three vectors. In line geometry, it tests coplanarity and calculates the distance between skew lines.

• Distance from a Point to a Line: The distance from a point with position vector $\mathbf{p}$ to

$$\mathbf{r}=\mathbf{a}+\lambda \mathbf{b}$$ is $$d=\frac{|(\mathbf{p}-\mathbf{a})\times \mathbf{b}|}{|\mathbf{b}|}.$$

• Opposite Direction Vectors: Replacing $(a,b,c)$ by $(-a,-b,-c)$ represents the same line, because the two direction vectors point in opposite directions along that line.

• Zero Direction Ratio: In Cartesian form, a zero direction ratio means that the corresponding coordinate remains constant. This case must be expressed using a separate coordinate equation rather than division by zero.

Worked Methods

1. Finding Direction Cosines from Direction Ratios

• Identify the direction ratios $a,b,c$.
• Calculate the magnitude

$$\sqrt{a^2+b^2+c^2}.$$

• Divide each direction ratio by this magnitude:

$$(l,m,n)=\frac{(a,b,c)}{\sqrt{a^2+b^2+c^2}}.$$

• Verify, if required, that

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

2. Writing a Line in Vector, Cartesian, and Parametric Forms

For a line through $(x_1,y_1,z_1)$ with direction ratios $a,b,c$:

• Write the position vector of the given point as

$$\mathbf{a}=x_1\mathbf{i}+y_1\mathbf{j}+z_1\mathbf{k}.$$

• Form the direction vector

$$\mathbf{b}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}.$$

• Write the vector equation:

$$\mathbf{r}=\mathbf{a}+\lambda \mathbf{b}.$$

• Write the parametric equations:

$$x=x_1+\lambda a,y=y_1+\lambda b,z=z_1+\lambda c.$$

• Write the Cartesian equation:

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

• If one direction ratio is zero, keep the corresponding coordinate constant and use a separate equation.

3. Finding the Line Through Two Points

For $$A(x_1,y_1,z_1),B(x_2,y_2,z_2):$$

• Subtract the coordinates of $A$ from those of $B$.
• Use

$$(x_2-x_1,y_2-y_1,z_2-z_1)$$ as direction ratios.

• If $\mathbf{a}$ and $\mathbf{b}$ are the position vectors of $A$ and $B$, write

$$\mathbf{r}=\mathbf{a}+\lambda (\mathbf{b}-\mathbf{a}).$$

• The distance $AB$, when required, is

$$AB=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}.$$

4. Finding the Angle Between Two Lines

For direction vectors ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$:

• Compute the dot product ${\mathbf{b}}_1\cdot {\mathbf{b}}_2$.
• Compute the magnitudes $|{\mathbf{b}}_1|$ and $|{\mathbf{b}}_2|$.
• Use

$$\cos \theta =\frac{|{\mathbf{b}}_1\cdot {\mathbf{b}}_2|}{|{\mathbf{b}}_1||{\mathbf{b}}_2|}.$$

• The absolute value gives the acute angle between the two lines.

5. Classifying Two Lines

For $$\mathbf{r}={\mathbf{a}}_1+\lambda {\mathbf{b}}_1,\mathbf{r}={\mathbf{a}}_2+\mu {\mathbf{b}}_2:$$

• Test parallelism:

$${\mathbf{b}}_1\times {\mathbf{b}}_2=0.$$

• If the direction vectors are parallel, determine whether the lines coincide or are distinct parallel lines by checking their points.
• If the directions are not parallel, test intersection by solving the parametric equations simultaneously.
• If a common point exists, the lines intersect.
• If no common point exists, test coplanarity:

$$({\mathbf{a}}_2-{\mathbf{a}}_1)\cdot ({\mathbf{b}}_1\times {\mathbf{b}}_2)=0.$$

• Non-parallel, non-intersecting lines that fail this coplanarity test are skew lines.

6. Finding the Shortest Distance Between Lines

First identify the line type.

• Intersecting lines:

1. Solve the parametric equations to find a common point. 2. Use $$d=0.$$

• Parallel lines:

1. Confirm $${\mathbf{b}}_1\times {\mathbf{b}}_2=0.$$ 2. Use a common direction vector $\mathbf{b}$. 3. Select one point from each line, with position vectors ${\mathbf{a}}_1,{\mathbf{a}}_2$. 4. Calculate $$d=\frac{|({\mathbf{a}}_2-{\mathbf{a}}_1)\times \mathbf{b}|}{|\mathbf{b}|}.$$

• Skew lines:

1. Confirm that the lines are non-parallel and non-intersecting. 2. Form the common perpendicular direction: $${\mathbf{b}}_1\times {\mathbf{b}}_2.$$ 3. Form the joining vector: $${\mathbf{a}}_2-{\mathbf{a}}_1.$$ 4. Project the joining vector onto the unit common-perpendicular direction: $$d=\frac{|({\mathbf{a}}_2-{\mathbf{a}}_1)\cdot ({\mathbf{b}}_1\times {\mathbf{b}}_2)|}{|{\mathbf{b}}_1\times {\mathbf{b}}_2|}.$$

7. Testing Coplanarity

• Take one point from each line, with position vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$.
• Identify the direction vectors ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$.
• Calculate the scalar triple product:

$$({\mathbf{a}}_2-{\mathbf{a}}_1)\cdot ({\mathbf{b}}_1\times {\mathbf{b}}_2).$$

• If the result is zero, the lines are coplanar.
• If the result is non-zero, the lines are not coplanar; when they are also non-parallel, they are skew.

Where It Goes Wrong

• Direction ratios are treated as unique rather than proportional. Any non-zero scalar multiple gives the same line direction, and changing $(a,b,c)$ to $(-a,-b,-c)$ represents the same line.

• Direction ratios are confused with direction cosines. Direction cosines must be normalized and satisfy

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

• The absolute value is omitted when finding the acute angle between two lines or the shortest distance between skew lines.

• The shortest-distance formula is selected before classifying the lines. Intersecting, parallel, and skew lines require different procedures and formulas.

• The coplanarity condition is omitted or applied incorrectly. For the two line equations, the required condition is

$$({\mathbf{a}}_2-{\mathbf{a}}_1)\cdot ({\mathbf{b}}_1\times {\mathbf{b}}_2)=0.$$

• A zero direction ratio is substituted into Cartesian form, producing division by zero. The corresponding coordinate must instead be written as constant.

What Gets Asked

This material supports questions requiring students to:

• Find direction cosines from given direction ratios and verify $l^2+m^2+n^2=1$.
• Determine the position vector of a point in three-dimensional coordinates.
• Write a line in vector, Cartesian, and parametric forms.
• Find the equation of the line joining two given points.
• Calculate the distance between two points.
• Find the acute angle between two lines using direction ratios or direction vectors.
• Determine whether two lines are parallel or perpendicular.
• Test whether two lines are coplanar.
• Identify skew lines from their geometric and algebraic properties.
• Find the shortest distance between intersecting, parallel, and skew lines.
• Determine the direction of the common perpendicular using a cross product.
• Find the distance from a point to a line.
• Interpret a zero direction ratio and represent the resulting constant coordinate correctly.