Learning these TS Inter 1st Year Maths 1B Formulas Chapter 7 The Plane will help students to solve mathematical problems quickly.

## TS Inter 1st Year Maths 1B The Plane Formulas

→ A plane is a proper subset of R’* which has atleast three non-collinear points and is such that for any two points in it. the line joining them also lies in it.

→ The general equation of a plane in the first degree equation in x, y, z given by ax + by + cz + d = 0. the coefficients a, b, c represent direction ratios of normal to the plane.

→ The equation of a plane passing through (x_{1}, y_{1}, z_{1}) and perpendicular to the line with direction ratios a, b, c is a (x – x_{1}) + b (y – y_{1}) + c (z – z_{1}) = 0.

→ Normal form of the plane is lx + my + nz – p where /. rn. n are direction cosine’s of normal and p is the perpendicular distance from origin to the plane.

→ The perpendicular distance from (0, 0, 0) to ax + by + cz t d = 0 is \(\frac{|d|}{\sqrt{a^2+b^2+c^2}\)

→ The perpendicular distance from A (x_{1}, y_{1}, z_{1}) to the plane ax + by + cz + d = 0 is \(\frac{\left|a x_1+b y_1+c z_1+d\right|}{\sqrt{a^2+b^2+c^2}}\)

→ The distance between parallel planes ax + by + cz + d_{1} = 0 and ax + by + cz + d_{2} = 0 is \(\frac{\left|d_1-d_2\right|}{\sqrt{a^2+b^2+c^2}}\)

→ The equation of plane with x. y. z intercepts a. b. c is \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\) = 1.

→ The equation of the plane passing through 3 non-collinear points A (x_{1}, y_{1} z_{1}). B (x_{2}, y_{2}, z_{2}) and C (x_{3}, y_{3} z_{3}) is \(\left|\begin{array}{ccc}

x-x_1 & y-y_1 & z-z_1 \\

x_2-x_1 & y_2-y_1 & z_2-z_1 \\

x_3-x_1 & y_3-y_1 & z_3-z_1

\end{array}\right|\) = 0

→ If θ is the angle between planes a_{1}x + b_{1}y + c_{1}z – d_{1} = 0 and a_{2}x + b_{2}y + c_{2}z + d_{2} = 0 then cos θ = \(\)

→ The planes a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 and a_{2}x + b_{2}y – c_{2}z + d = 0 are parallel if \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\) and perpendicular if a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0.