### Three Dimensional Geometry

**How to Remember JEE and CBSE Board Exam Concepts - Three Dimensional Geometry**

**Lines and Planes**

**Equations of lines and planes in space:**

- Use parametric equations to represent lines.
- Use the vector equation of a plane to represent planes.

**Angle between two lines and between a line and a plane:**

- Use the dot product of two vectors to find the angle between them.
- Use the angle between a line and a plane to find the shortest distance from the line to the plane.

**Distance from a point to a line and from a point to a plane:**

- Use the perpendicular distance formula to find the distance from a point to a line or a plane.

**Spheres**

**Equation of a sphere:**

- Use the standard equation of a sphere: (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2, where (h, k, l) is the center of the sphere and r is the radius.

**Center and radius of a sphere:**

- Identify the center and radius of a sphere from its equation.

**Intersection of a sphere with a line and with a plane:**

- Find the points of intersection of a sphere with a line by substituting the parametric equations of the line into the equation of the sphere.
- Find the points of intersection of a sphere with a plane by substituting the equation of the plane into the equation of the sphere.

**Tangent planes to a sphere:**

- Find the equation of a tangent plane to a sphere at a given point by using the gradient of the sphere at that point.

**Cones and Cylinders**

**Equations of cones and cylinders:**

- Use the standard equations of cones and cylinders:
- Cone: (x - h)^2 + (y - k)^2 = z^2/a^2
- Right circular cone: x^2 + y^2 = (z-k)^2/a^2
- Cylinder: x^2 + y^2 = r^2

**Right circular cones and cylinders:**

- Identify a cone or cylinder as right circular if its sides are perpendicular to its base.

**Slant height and surface area of a cone:**

- The slant height of a cone is the distance from the vertex of the cone to the base.
- The surface area of a cone is equal to the sum of the areas of the base and the sides.

**Curved surface area and volume of a cylinder:**

- The curved surface area of a cylinder is equal to 2πRh, where R is the radius of the base and H is the height of the cylinder.
- The volume of a cylinder is equal to πR^2H.

**Vectors**

**Dot and cross products of two vectors:**

- The dot product of two vectors is a scalar quantity that represents the projection of one vector onto the other.
- The cross product of two vectors is a vector that is perpendicular to both vectors and has a magnitude equal to the area of the parallelogram formed by the two vectors.

**Scalar triple product:**

- The scalar triple product of three vectors is a scalar quantity that represents the volume of the parallelepiped formed by the three vectors.

**Vector equations of lines and planes:**

- Use vector equations to represent lines and planes in three-dimensional space.

**Coordinate Geometry of Three Dimensions**

**Distance between two points:**

- Use the distance formula to find the distance between two points in three-dimensional space.

**Direction cosines and direction ratios:**

- The direction cosines of a vector are the cosines of the angles that the vector makes with the positive x-, y-, and z-axes.
- The direction ratios of a vector are the ratios of the components of the vector to the magnitude of the vector.

**Equations of a sphere, plane, and line:**

- Use the standard equations to represent spheres, planes, and lines in three-dimensional space.

**Skew lines:**

- Two lines are skew if they do not intersect and are not parallel.

**Coplanar lines:**

- Three or more lines are coplanar if they lie in the same plane.

**Applications of Three Dimensional Geometry**

**Finding the shortest distance between two points:**

- Use the distance formula to find the shortest distance between two points.

**Finding the volume of a solid:**

- Use the appropriate formula to find the volume of a solid, such as the volume of a sphere, cone, cylinder, or prism.

**Finding the surface area of a solid:**

- Use the appropriate formula to find the surface area of a solid, such as the surface area of a sphere, cone, cylinder, or prism.

**Solving problems related to vectors:**

- Use vectors to solve problems involving forces, moments, and other vector quantities.