Three Dimensional Geometry Chapter

Three Dimensional Geometry

Coordinate Axes in 3D Space

• Three mutually perpendicular lines in 3D space, intersecting at a point called the origin (0, 0, 0)
• X-axis, Y-axis, and Z-axis

Coordinates of a Point in 3D Space

• A point in 3D space is represented by an ordered triplet (x, y, z)
• x, y, and z are the distances of the point from the YZ-plane, ZX-plane, and XY-plane respectively

Distance Formula in 3D Space

• Distance between two points P(x1, y1, z1) and Q(x2, y2, z2) is given by:

Distance = √[(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2]

Section Formula in 3D Space

• The coordinates of a point that divides the line segment joining two points P(x1, y1, z1) and Q(x2, y2, z2) in the ratio m:n is given by:

(x, y, z) = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n), (mz2 + nz1)/(m+n))

Direction Cosines and Direction Ratios

• Direction cosines of a line are the cosines of the angles made by the line with the positive directions of the X, Y, and Z axes
• Direction ratios of a line are the numbers proportional to the direction cosines
• If l, m, and n are the direction cosines, then a, b, and c are the direction ratios if:

a = kl, b = km, c = kn

Vector and Cartesian Equation of a Line

• Vector equation of a line passing through a point with position vector a and parallel to vector b is:

r = a + λb

• Cartesian equation of a line passing through a point (x1, y1, z1) and having direction ratios a, b, and c is:

(x - x1)/a = (y - y1)/b = (z - z1)/c

Plane in 3D Space

• Vector equation of a plane passing through a point with position vector a and having normal vector n is:

(r - a) . n = 0

• Cartesian equation of a plane is of the form:

Ax + By + Cz + D = 0

• Normal vector to the plane is (A, B, C)

Coordinate Axes in 3D Space

• Three mutually perpendicular lines in 3D space intersect at the origin (0, 0, 0)
• The three axes are the X-axis, Y-axis, and Z-axis

Coordinates of a Point in 3D Space

• A point in 3D space is represented by an ordered triplet (x, y, z)
• x, y, and z are the distances of the point from the YZ-plane, ZX-plane, and XY-plane respectively

Distance Formula in 3D Space

• The distance between two points P(x1, y1, z1) and Q(x2, y2, z2) is given by: √[(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2]

Section Formula in 3D Space

• The coordinates of a point that divides the line segment joining two points P(x1, y1, z1) and Q(x2, y2, z2) in the ratio m:n is given by: (x, y, z) = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n), (mz2 + nz1)/(m+n))

Direction Cosines and Direction Ratios

• Direction cosines of a line are the cosines of the angles made by the line with the positive directions of the X, Y, and Z axes
• Direction ratios of a line are the numbers proportional to the direction cosines
• If l, m, and n are the direction cosines, then a, b, and c are the direction ratios if a = kl, b = km, c = kn

Vector and Cartesian Equation of a Line

• Vector equation of a line passing through a point with position vector a and parallel to vector b is: r = a + λb
• Cartesian equation of a line passing through a point (x1, y1, z1) and having direction ratios a, b, and c is: (x - x1)/a = (y - y1)/b = (z - z1)/c

Plane in 3D Space

• Vector equation of a plane passing through a point with position vector a and having normal vector n is: (r - a).n = 0
• Cartesian equation of a plane is of the form: Ax + By + Cz + D = 0
• Normal vector to the plane is (A, B, C)

Learn about coordinate axes, coordinates of a point, and distance formula in 3D space. Explore the concepts of X, Y, and Z axes, and how to represent points in 3D space.