Three Dimensional Geometry Class 12 Chapter Notes

Key Concepts to Remember

• Direction Cosines (l, m, n) are cosines of angles a line makes with x, y, z axes. Always $l^2 + m^2 + n^2 = 1$.
• Direction Ratios (a, b, c) are numbers proportional to direction cosines. If (a, b, c) are DRs, then (a/√(a²+b²+c²), b/√(a²+b²+c²), c/√(a²+b²+c²)) are DCs.
• A line in space can be represented in vector form $ \vec{r} = \vec{a} + \lambda \vec{b} $ or Cartesian form $(x-x_1)/a = (y-y_1)/b = (z-z_1)/c$.
• The shortest distance between two skew lines is given by $| (\vec{b_1} \times \vec{b_2}) \cdot (\vec{a_2} - \vec{a_1}) | / | \vec{b_1} \times \vec{b_2} |$.
• A plane in space can be represented in vector form $ \vec{r} \cdot \vec{n} = d $ or Cartesian form $Ax + By + Cz = D$.
• The angle between two planes with normal vectors $ \vec{n_1} $ and $ \vec{n_2} $ is $ \theta = \cos^{-1} ( |\vec{n_1} \cdot \vec{n_2}| / (|\vec{n_1}| |\vec{n_2}|) ) $. For a line and a plane, use $ \sin \theta = ( |\vec{b} \cdot \vec{n}| / (|\vec{b}| |\vec{n}|) ) $.
• Coplanarity of two lines occurs if the shortest distance between them is zero, i.e., $ (\vec{b_1} \times \vec{b_2}) \cdot (\vec{a_2} - \vec{a_1}) = 0 $.
• The distance of a point from a plane $Ax + By + Cz + D = 0$ is $ |Ax_1 + By_1 + Cz_1 + D| / \sqrt{A^2 + B^2 + C^2} $.

Key Definitions

Direction Cosines (DCs)

The cosines of the angles made by a line in space with the positive directions of the coordinate axes (x, y, z). Denoted by l, m, n. Property: $l^2 + m^2 + n^2 = 1$.

Direction Ratios (DRs)

Any three numbers a, b, c which are proportional to the direction cosines of a line. If (a, b, c) are DRs, then $(a/\sqrt{a^2+b^2+c^2}, b/\sqrt{a^2+b^2+c^2}, c/\sqrt{a^2+b^2+c^2})$ are its DCs.

Skew Lines

Two lines in three-dimensional space that are neither parallel nor intersecting. They lie in different planes.

Coplanar Lines

Two lines that lie in the same plane. They either intersect or are parallel.

Normal to a Plane

A vector that is perpendicular to every vector lying in the plane. It defines the orientation of the plane in space.

Intercept Form of a Plane

The equation of a plane in the form $x/a + y/b + z/c = 1$, where a, b, c are the x, y, z-intercepts respectively.

Line of Intersection of Two Planes

The common line shared by two non-parallel planes. Its direction vector is parallel to the cross product of the normal vectors of the two planes.

Understanding Lines and Planes in 3D Space

Three-dimensional geometry extends our understanding of shapes and positions from a 2D plane to a 3D space. The fundamental elements we study are points, lines, and planes. A point in 3D is uniquely identified by its coordinates $(x, y, z)$.

A line in space is essentially a path that extends infinitely in two opposite directions. To define a line, we generally need two pieces of information: either a point on the line and its direction, or two distinct points on the line. The direction of a line is crucial and is specified using direction ratios (DRs) or direction cosines (DCs). If a line passes through a point $A(\vec{a})$ and is parallel to a vector $ \vec{b} $, its vector equation is $ \vec{r} = \vec{a} + \lambda \vec{b} $. In Cartesian form, if $A = (x_1, y_1, z_1)$ and $ \vec{b} = a \hat{i} + b \hat{j} + c \hat{k} $, the equation becomes $(x-x_1)/a = (y-y_1)/b = (z-z_1)/c$.

A plane in space is a flat, two-dimensional surface that extends infinitely. To uniquely define a plane, we often need a point on the plane and a vector normal (perpendicular) to it. If a plane passes through a point $A(\vec{a})$ and its normal vector is $ \vec{n} $, its vector equation is $ (\vec{r} - \vec{a}) \cdot \vec{n} = 0 $, or $ \vec{r} \cdot \vec{n} = \vec{a} \cdot \vec{n} $. In Cartesian form, if $ \vec{n} = A \hat{i} + B \hat{j} + C \hat{k} $, the general equation is $Ax + By + Cz + D = 0$. Understanding these forms is key to solving problems involving angles, distances, and intersections between lines and planes. Concepts like shortest distance between skew lines and coplanarity of lines are direct applications of vector algebra in 3D space, which you studied in the previous chapter. Always remember to convert between vector and Cartesian forms as needed, depending on the problem's context.

Comparing Lines and Planes in Different Forms

Worked Examples

• Example 1: Equation of a Line Q: Find the vector equation of a line passing through point $(1,2,-4)$ and parallel to vector $2\hat{i} - 3\hat{j} + 5\hat{k}$. A: Given point $ \vec{a} = \hat{i} + 2\hat{j} - 4\hat{k} $ and parallel vector $ \vec{b} = 2\hat{i} - 3\hat{j} + 5\hat{k} $. The vector equation of the line is $ \vec{r} = \vec{a} + \lambda \vec{b} $. Substituting the values, we get $ \vec{r} = (\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda (2\hat{i} - 3\hat{j} + 5\hat{k}) $.
• Example 2: Angle between two planes Q: Find the angle between the planes $2x + y - 2z = 5$ and $3x - 6y - 2z = 7$. A: The normal vectors are $ \vec{n_1} = 2\hat{i} + \hat{j} - 2\hat{k} $ and $ \vec{n_2} = 3\hat{i} - 6\hat{j} - 2\hat{k} $. $ |\vec{n_1}| = \sqrt{2^2 + 1^2 + (-2)^2} = \sqrt{4+1+4} = \sqrt{9} = 3 $. $ |\vec{n_2}| = \sqrt{3^2 + (-6)^2 + (-2)^2} = \sqrt{9+36+4} = \sqrt{49} = 7 $. $ \vec{n_1} \cdot \vec{n_2} = (2)(3) + (1)(-6) + (-2)(-2) = 6 - 6 + 4 = 4 $. $ \cos \theta = |\vec{n_1} \cdot \vec{n_2}| / (|\vec{n_1}| |\vec{n_2}|) = 4 / (3 \cdot 7) = 4/21 $. So, $ \theta = \cos^{-1}(4/21) $.

Exam Traps & Scoring Tips

1. Vector vs. Cartesian Forms: Be proficient in converting between vector and Cartesian forms for both lines and planes. Examiners often switch formats to test your understanding. Always read the question carefully to see which form is required for the answer.
2. Shortest Distance: For parallel lines, the shortest distance formula is different from skew lines. Don't confuse them! Skew lines: $ | (\vec{b_1} \times \vec{b_2}) \cdot (\vec{a_2} - \vec{a_1}) | / | \vec{b_1} \times \vec{b_2} | $. Parallel lines: $ | \vec{b} \times (\vec{a_2} - \vec{a_1}) | / | \vec{b} | $.
3. Angles: Remember that the angle between two lines uses $ \cos \theta = |\vec{b_1} \cdot \vec{b_2}| / (|\vec{b_1}| |\vec{b_2}|) $, and between two planes uses $ \cos \theta = |\vec{n_1} \cdot \vec{n_2}| / (|\vec{n_1}| |\vec{n_2}|) $. However, for the angle between a line and a plane, it's $ \sin \theta = ( |\vec{b} \cdot \vec{n}| / (|\vec{b}| |\vec{n}|) ) $. A common mistake is using cosine for line-plane angle.
4. Equation of Plane through Intersection: The equation of a plane passing through the intersection of two planes $P_1: \vec{r} \cdot \vec{n_1} = d_1$ and $P_2: \vec{r} \cdot \vec{n_2} = d_2$ is $ \vec{r} \cdot (\vec{n_1} + \lambda \vec{n_2}) = d_1 + \lambda d_2 $. Don't forget the scalar $ \lambda $ and ensure you use the correct form based on the given problem.

Practice Questions with Solutions

• Q1: If a line has direction ratios (2, -1, 2), what are its direction cosines? A1: DCs are $(2/3, -1/3, 2/3)$.
• Q2: Write the Cartesian equation of a line passing through $(3,0,-1)$ and parallel to vector $3\hat{i} + 2\hat{j} - 5\hat{k}$. A2: $(x-3)/3 = (y-0)/2 = (z-(-1))/-5 \Rightarrow (x-3)/3 = y/2 = (z+1)/-5$.
• Q3: What is the condition for two lines $ \vec{r} = \vec{a_1} + \lambda \vec{b_1} $ and $ \vec{r} = \vec{a_2} + \mu \vec{b_2} $ to be coplanar? A3: The lines are coplanar if $ (\vec{a_2} - \vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2}) = 0 $.
• Q4: If the normal to a plane is $ \hat{i} + 2\hat{j} - 3\hat{k} $ and it passes through the origin, what is its vector equation? A4: $ \vec{r} \cdot (\hat{i} + 2\hat{j} - 3\hat{k}) = 0 $.

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