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 (x1, y1, z1) and perpendicular to the line with direction ratios a, b, c is a (x – x1) + b (y – y1) + c (z – z1) = 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 (x1, y1, z1) 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 + d1 = 0 and ax + by + cz + d2 = 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 (x1, y1 z1). B (x2, y2, z2) and C (x3, y3 z3) 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 a1x + b1y + c1z – d1 = 0 and a2x + b2y + c2z + d2 = 0 then cos θ = \(\)
→ The planes a1x + b1y + c1z + d1 = 0 and a2x + b2y – c2z + d = 0 are parallel if \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\) and perpendicular if a1a2 + b1b2 + c1c2 = 0.