Three-dimensional Geometry

The One Thing

Three-dimensional coordinate geometry represents points, lines, and planes using coordinates and vectors. Dot products determine angles and perpendicularity, while cross products construct normal vectors and calculate shortest distances.

Definitions and Results

• Coordinate Point: A point in three-dimensional space is represented as $(x,y,z)$, where $x$, $y$, and $z$ are its coordinates along three mutually perpendicular axes.

• Position Vector: The position vector of $P(x,y,z)$ with respect to the origin is

$$\mathbf{r}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}.$$

• Direction Ratios: Three numbers proportional to the changes in $x$, $y$, and $z$ along a line, commonly written as $(a,b,c)$.

• Direction Cosines: If a line makes angles with the positive $x$-, $y$-, and $z$-axes whose cosines are $l,m,n$, then

$$l=\frac{a}{\sqrt{a^2+b^2+c^2}},m=\frac{b}{\sqrt{a^2+b^2+c^2}},n=\frac{c}{\sqrt{a^2+b^2+c^2}},$$ and $$l^2+m^2+n^2=1.$$

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

$$\mathbf{r}=\mathbf{a}+\lambda \mathbf{b},$$ where $\lambda$ is a real parameter. Through points with position vectors $\mathbf{a}$ and $\mathbf{b}$, the equation is $$\mathbf{r}=\mathbf{a}+\lambda (\mathbf{b}-\mathbf{a}).$$

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

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

• Equation of a Plane: A plane with normal vector $\mathbf{n}$ passing through position vector $\mathbf{a}$ is

$$(\mathbf{r}-\mathbf{a})\cdot \mathbf{n}=0.$$

• Cartesian Equation of a Plane: The general equation is

$$ax+by+cz+d=0,$$ where $(a,b,c)$ are direction ratios of a normal to the plane. The coefficients $a,b,c$ must not all be zero.

• Normal Vector: A vector perpendicular to every line lying in a plane. For

$$ax+by+cz+d=0,$$ a normal vector is $$a\mathbf{i}+b\mathbf{j}+c\mathbf{k}.$$

• Dot Product: For vectors $\mathbf{a}$ and $\mathbf{b}$,

$$\mathbf{a}\cdot \mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos \theta =a_1{b}_1+a_2{b}_2+a_3{b}_3.$$

• Cross Product: The vector $\mathbf{a}\times \mathbf{b}$ is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$, and

$$|\mathbf{a}\times \mathbf{b}|=|\mathbf{a}||\mathbf{b}|\sin \theta .$$

• Angle Between Two Lines: For direction ratios $(a_1,b_1,c_1)$ and $(a_2,b_2,c_2)$,

$$\cos \theta =\frac{|a_1{a}_2+b_1{b}_2+c_1{c}_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}.$$

• Angle Between Two Planes: This is the angle between their normal vectors. If the normals have direction ratios $(a_1,b_1,c_1)$ and $(a_2,b_2,c_2)$, use the dot-product formula for $\cos \theta$.

• Angle Between a Line and a Plane: If the line has direction vector $\mathbf{d}$ and the plane has normal vector $\mathbf{n}$, then

$$\sin \theta =\frac{|\mathbf{d}\cdot \mathbf{n}|}{|\mathbf{d}||\mathbf{n}|},$$ where $\theta$ is the angle between the line and the plane.

• Skew Lines: Two lines in space that are neither parallel nor intersecting.

• Coplanar Lines: Two lines are coplanar if they lie in the same plane. For lines with direction vectors ${\mathbf{b}}_1,{\mathbf{b}}_2$ and joining vector ${\mathbf{a}}_2-{\mathbf{a}}_1$, they are coplanar when

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

• Shortest Distance Between Two 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 $$\frac{|({\mathbf{a}}_2-{\mathbf{a}}_1)\cdot ({\mathbf{b}}_1\times {\mathbf{b}}_2)|}{|{\mathbf{b}}_1\times {\mathbf{b}}_2|}.$$

• Distance of a Point from a Plane: The distance of $(x_1,y_1,z_1)$ from

$$ax+by+cz+d=0$$ is $$\frac{|a{x}_1+b{y}_1+c{z}_1+d|}{\sqrt{a^2+b^2+c^2}}.$$

• Distance Between Parallel Planes: For

$$ax+by+cz+d_1=0,ax+by+cz+d_2=0,$$ the distance is $$\frac{|d_1-d_2|}{\sqrt{a^2+b^2+c^2}}.$$

• Plane Through Three Non-Collinear Points: Find two vectors joining one point to the other two, take their cross product to obtain a normal vector, and use the point-normal equation.

• Parallel and Perpendicular Conditions:

- A line with direction vector $\mathbf{d}$ is parallel to a plane with normal vector $\mathbf{n}$ when $$\mathbf{d}\cdot \mathbf{n}=0.$$ - A line is perpendicular to a plane when its direction vector is parallel to the plane’s normal vector. - Two planes are parallel when their normal vectors are proportional. - Two planes are perpendicular when the dot product of their normal vectors is zero. - Two lines are parallel when their direction ratios are proportional. - Two lines are perpendicular when the dot product of their direction vectors is zero. - A line lies in a plane when one point of the line satisfies the plane equation and the line direction vector is perpendicular to the plane’s normal vector.

• Scalar Triple Product: The scalar triple product is

$$[\mathbf{a},\mathbf{b},\mathbf{c}]=\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c}).$$ It is zero when the three vectors are coplanar.

Worked Methods

1. Representing a Point and Its Position Vector

• Identify the coordinates $P(x,y,z)$.
• Write the position vector from the origin as

$$\mathbf{r}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}.$$

2. Finding Direction Cosines

• Start with direction ratios $(a,b,c)$.
• Calculate the magnitude

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

• Divide each direction ratio by this magnitude:

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

• Verify the required condition:

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

3. Forming the Equation of a Line

• Choose a point on the line with position vector $\mathbf{a}$.
• Determine a direction vector $\mathbf{b}$.
• Substitute into

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

• Alternatively, if the line passes through $(x_1,y_1,z_1)$ and has direction ratios $(a,b,c)$, use

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

• If two points have position vectors $\mathbf{a}$ and $\mathbf{b}$, use

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

4. Forming the Equation of a Plane

• Identify a point $(x_1,y_1,z_1)$ on the plane.
• Identify a normal vector $(a,b,c)$.
• Use the point-normal equation:

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

• Expand into the general form

$$ax+by+cz+d=0$$ where required.

5. Constructing a Plane Through Three Non-Collinear Points

• Let the points be $P,Q,R$.
• Form two joining vectors, such as

$$\overrightarrow{PQ}=Q-P,\overrightarrow{PR}=R-P.$$

• Calculate

$$\overrightarrow{PQ}\times \overrightarrow{PR}.$$

• Use the resulting vector as a normal vector.
• Substitute one of the points into

$$(\mathbf{r}-\mathbf{a})\cdot \mathbf{n}=0$$ or $$a(x-x_1)+b(y-y_1)+c(z-z_1)=0.$$

6. Finding Angles

• For two lines, identify their direction vectors or direction ratios.
• Calculate their dot product.
• Substitute into

$$\cos \theta =\frac{|a_1{a}_2+b_1{b}_2+c_1{c}_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}.$$

• For two planes, use their normal vectors in the same dot-product formula.
• For a line and a plane, use the line direction vector $\mathbf{d}$ and plane normal $\mathbf{n}$:

$$\sin \theta =\frac{|\mathbf{d}\cdot \mathbf{n}|}{|\mathbf{d}||\mathbf{n}|}.$$

7. Testing Relative Positions

• Compare line direction vectors:

- proportional vectors indicate parallel lines; - zero dot product indicates perpendicular lines.

• Compare plane normal vectors:

- proportional normals indicate parallel planes; - zero dot product indicates perpendicular planes.

• To determine whether a line is parallel to a plane, check

$$\mathbf{d}\cdot \mathbf{n}=0.$$

• To determine whether a line lies in a plane, verify both that one point satisfies the plane equation and that

$$\mathbf{d}\cdot \mathbf{n}=0.$$

• To test whether lines are coplanar, calculate

$$({\mathbf{a}}_2-{\mathbf{a}}_1)\cdot ({\mathbf{b}}_1\times {\mathbf{b}}_2).$$ A result of zero indicates coplanarity; otherwise, the lines are skew.

8. Calculating Distances

• For a point and a plane, substitute the point into

$$\frac{|a{x}_1+b{y}_1+c{z}_1+d|}{\sqrt{a^2+b^2+c^2}}.$$

• For parallel planes, use

$$\frac{|d_1-d_2|}{\sqrt{a^2+b^2+c^2}}.$$

• For skew lines, form ${\mathbf{b}}_1\times {\mathbf{b}}_2$, then use

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

9. Testing Whether a Point Lies on a Line or Plane

• Substitute the point’s coordinates into the relevant Cartesian line or plane equation.
• If the equation is satisfied, the point lies on the line or plane.
• For a line in a plane, also check that one point of the line satisfies the plane equation and that the line direction vector is perpendicular to the plane’s normal vector.

Where It Goes Wrong

• Direction ratios are not necessarily unit vectors; direction cosines require division by $\sqrt{a^2+b^2+c^2}$.
• The condition $l^2+m^2+n^2=1$ must be satisfied by direction cosines.
• A plane equation requires a normal vector; in $ax+by+cz+d=0$, $(a,b,c)$ must not all be zero.
• The angle between a line and a plane uses

$$\sin \theta =\frac{|\mathbf{d}\cdot \mathbf{n}|}{|\mathbf{d}||\mathbf{n}|},$$ not the direct line-angle cosine formula.

• A line being parallel to a plane requires $\mathbf{d}\cdot \mathbf{n}=0$, but a line lies in the plane only when one point also satisfies the plane equation.
• Lines that are neither parallel nor intersecting are skew; the scalar triple product distinguishes skew lines from coplanar lines.
• For a plane through three non-collinear points, the cross product must be formed from two joining vectors, and the resulting vector is used as the normal.

What Gets Asked

This material supports questions requiring students to:

• Find position vectors, direction ratios, and direction cosines.
• Form vector and Cartesian equations of lines.
• Form vector and Cartesian equations of planes.
• Find a plane through three non-collinear points.
• Calculate dot products, cross products, and scalar triple products.
• Determine angles between two lines, two planes, or a line and a plane.
• Test whether lines or planes are parallel, perpendicular, intersecting, coplanar, or skew.
• Determine whether a line lies in or is parallel to a plane.
• Test whether a point lies on a line or plane by substitution.
• Calculate the distance from a point to a plane, between parallel planes, and between two skew lines.
• Justify geometrical relationships by converting them into algebraic conditions involving direction vectors, normal vectors, dot products, and cross products.