The Planes

The general equation of a plane

The general equation of a plane in three-dimensional space is given by \( ax + by + cz + d = 0 \), where \( a, b, c \) are direction ratios of the nor

If a plane passes through

If a plane passes through a point \( (x_1, y_1, z_1) \), its equation is \( a(x - x_1) + b(y - y_1) + c(z - z_1) = 0 \), where \( a, b, c \) are the d

When a plane makes intercepts \(

When a plane makes intercepts \( a, b, c \) on the x, y, and z-axes respectively, its equation is \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \).

The normal form of a plane

The normal form of a plane is \( lx + my + nz = p \), where \( l, m, n \) are the direction cosines of the normal to the plane, and \( p \) is the per

The angle \( \theta \) between

The angle \( \theta \) between two planes is the angle between their normals. It is given by \( \cos\theta = \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a