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.