Use these Inter 1st Year Maths 1B Formulas PDF Chapter 7 The Plane to solve questions creatively.

## Intermediate 1st Year Maths 1B The Plane Formulas

Definition: If F(x, y, z) is a polynomial of degree one, then the surface represented by F(x, y, z).= 0 is called a first degree

→ If A, B, C are three non-collinear points in space, then there exists exactly only one plane through A,

→ If a plane contains two points A and B, then the plane also contains the line \(\overleftrightarrow{A B}\).

→ If two planes intersect, then their intersection is a line.

→ If a plane π and a line L are perpendicular at a point P, then π contains every line that passes through P and is perpendicular to L.

→ Equation of a plane in the normal form is lx + my + nz = p. Hence (l, m, n) are D.C’s of the normal to the plane, and p( > 0) is the perpendicular distance of the plane from the origin.

→ If (a, b, c) & (0,0,0), then the equation ax + by + cz + d = 0 is called the general form of the equation of a plane.

→ Equation of the plane n which contains the point A(x_{0}, y_{0}, z_{0}) and perpendicular to the line L with direction ratios (a, b, c) is a(x – xQ) + b(y – y0) + c(z – z ) = 0.

→ Equation of the plane passing through three non-collinear point 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_{i}

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

→ Equation of the plane in the intercept form is \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\) = 1.

→ The angle between 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_{2} = 0 is cos^{-1} \(\frac{\left|a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}\right|}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\)

→ The perpendicular distance of the plance ax + by + cz + d = 0 from the point P(x_{0} , y_{0} , z_{0}) is \(\frac{\left|a x_{0}+b y_{0}+c z_{0}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}\)

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

**Planes:**

Definition:

A surface in space is said to be a plane surface or a plane if all the points of the straight line joining any two points of the surface lie on the surface.

Theorem:

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

Theorem:

The equation of the plane passing through a point (x_{1}, y_{1}, z_{1}) is a( x – x_{1})+b( y – y_{1})+c (z – z_{1}) = 0 where a,b,c are constants.

Theorem:

The equation of the plane containing three points (x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}), (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

**Normal Form of A Plane:**

Theorem:

The equation of the plane which is at a distance of p from the origin and whose normal has the direction cosines (l, m, n) is lx + my + nz = p (or) x cos α+ y cos β + z cos γ = p.

Note:

Equation of the plane through the origin is lx + my + nz = 0

Note: The equation of the plane ax +by+cz+d=0 in the normal form is

where Σa^{2} = a^{2} + b^{2} + c^{2}

**Perpendicular Distance From A Point to A Plane:**

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

Theorem

The perpendicular distance from P(x1, y1, z1) to the plane ax + by + cz + d = 0 is \(\frac{\left|a x_{1}+b y_{1}+c z_{1}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}\)

Theorem

Intercept form of the plane

The equation of the plane having a,b,c as x, y, z- intercepts respectively is \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\) = 1

Theorem

The intercepts of the plane ax + by + cz + d = 0 are respectively \(\frac{-d}{a}, \frac{-d}{b}, \frac{-d}{c}\)

Angle Between Two Planes:

Definition: The angle between the normals to two planes is called the angle between the planes.

Theorem:

If θ is the angle between the planes a_{1}x + b_{1} y + c_{1}z + d_{1} = 0, a_{2}x + b_{2}y + c_{2}z + d_{2} = 0 then cos θ = \(\frac{a_{1} a_{2}+b_{1} b_{1}+c_{1} c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\)

Note:

- If θ is acute then cos θ = \(\frac{a_{1} a_{2}+b_{1} b_{1}+c_{1} c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}} \mid\)
- The planes a
_{1}x + b_{1}y + c_{1}z + d_{1}= 0, a_{2}x + b_{2}y + c_{2}z + d_{2}= 0 are- parallel iff \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)
- Perpendicular iff . a
_{1}a_{2}+ b_{1}b_{2}+ c_{1}c_{2}= 0

- The given planes are perpendicular

⇔ θ = 90° ⇔ cos θ = 0 ⇒ a_{1}a_{2}+ b_{1}b_{2}+ c_{1}c_{2}= 0

Theorem:

The equation of the plane parallel to the plane ax + by + cz + d = 0 is ax + by + cz + k = 0 where k is a constant.

Theorem:

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