The plane 2x - (1 + a)y + 3az = 0 passes through the intersection of the planes

$$2x – y = 0$$ and $$y + 3z = 0$$

$$2x – y = 0$$ and $$y - 3z = 0$$

The plane 2x - (1 + a)y + 3az = 0 passes through the intersection of the planes

$$2x – y = 0 and y + 3z = 0$$

$$2x – y = 0 and y - 3z = 0$$

Three-dimensional geometry plays a major role as a lot of questions are included in the JEE exam. Here is a list of all the three-dimensional geometry formulas which will help students to go through and revise them quickly before the exam.

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Three-dimensional Geometry Formulas

1. Vector representation of a point: Position vector of a point P(x, y, z) is

\begin{array}{l} x \hat{i} + y \hat{j} + z \hat{k} \end{array}

2. Distance formula:

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

\begin{array}{l} P Q = \sqrt{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2} + (z_{1} - z_{2})^{2}} \end{array}

\begin{array}{l} A B = | \vec{O B} - \vec{O A} | \end{array}

3. Distance of P from coordinate axes:

\begin{array}{l} P A = \sqrt{(y^{2} + z^{2})} \end{array}

\begin{array}{l} P B = \sqrt{(z^{2} + x^{2})} \end{array}

\begin{array}{l} P C = \sqrt{(x^{2} + y^{2})} \end{array}

4. Section Formula:

\begin{array}{l} x = \frac{m x_{2} + n x_{1}}{m + n} \end{array}

\begin{array}{l} y = \frac{m y_{2} + n y_{1}}{m + n} \end{array}

\begin{array}{l} z = \frac{m z_{2} + n z_{1}}{m + n} \end{array}

Midpoint:

\begin{array}{l} x = \frac{x_{1} + x_{2}}{2} \end{array}

\begin{array}{l} y = \frac{y_{1} + y_{2}}{2} \end{array}

\begin{array}{l} z = \frac{z_{1} + z_{2}}{2} \end{array}

5. Centroid of a triangle:

\begin{array}{l} G = (\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3}, \frac{z_{1} + z_{2} + z_{3}}{3}) \end{array}

6. Incentre of triangle ABC:

\begin{array}{l} (\frac{a x_{1} + b x_{2} + c x_{3}}{a + b + c}, \frac{a y_{1} + b y_{2} + c y_{3}}{a + b + c}, \frac{a z_{1} + b z_{2} + c z_{3}}{a + b + c}) \end{array}

7. Centroid of a tetrahedron:

\begin{array}{l} (\frac{x_{1} + x_{2} + x_{3} + x_{4}}{4}, \frac{y_{1} + y_{2} + y_{3} + y_{4}}{4}, \frac{z_{1} + z_{2} + z_{3} + z_{4}}{4}) \end{array}

8. Direction cosines and direction ratios:

(i) Direction cosines: let α, β, γ be the angles which a directed line makes with the positive directions of the axes of x, y and z respectively, then cos α, cos β and cos γ are called the direction cosines of the line. The direction cosines are usually denoted by (l, m, n).

Therefore, l = cos α, m = cos β, n = cos γ.

(ii) l²+m²+n² = 1

(iii) If a, b, c are the direction ratios of any line L then

\begin{array}{l} a \hat{i} + b \hat{j} + c \hat{k} \end{array}

will be a vector parallel to the line L.

(iv) If l, m, and n are the direction cosines of any line L, then

\begin{array}{l} l \hat{i} + m \hat{j} + n \hat{k} \end{array}

is a unit vector parallel to the line L.

(v) If l, m, n be the direction cosines and a, b, c be the direction ratios of a vector, then

\begin{array}{l} l = \frac{a}{\sqrt{a^{2} + b^{2} + c^{2}}} \end{array}

\begin{array}{l} m = \frac{b}{\sqrt{a^{2} + b^{2} + c^{2}}} \end{array}

\begin{array}{l} n = \frac{c}{\sqrt{a^{2} + b^{2} + c^{2}}} \end{array}

\begin{array}{l} l = \frac{- a}{\sqrt{a^{2} + b^{2} + c^{2}}} \end{array}

\begin{array}{l} m = \frac{- b}{\sqrt{a^{2} + b^{2} + c^{2}}} \end{array}

\begin{array}{l} n = \frac{- c}{\sqrt{a^{2} + b^{2} + c^{2}}} \end{array}

(vi) If OP = r, the direction cosines of OP are l, m, n then the coordinates of P are (lr, mr, nr).

If the direction cosines of the line AB are l, m, n, |AB| = r and the coordinates of A is (x₁, y₁, z₁) then the coordinates of B are given as (x₁+ rl, y₁+ rm, z₁+ rn).

(vii) If the coordinates P and Q are (x₁, y₁, z₁) and (x₂, y₂, z₂) then the direction ratios of line PQ are a = x₂– x₁ , b = y₂– y₁ and c = z₂– z₁and the direction cosines of line PQ are:

\begin{array}{l} l = \frac{x_{2} - x_{1}}{| P Q |}, m = \frac{y_{2} - y_{1}}{| P Q |} n = \frac{z_{2} - z_{1}}{| P Q |} \end{array}

(vii) Direction cosines of the x-axis is (1, 0, 0).

Direction cosines of the y-axis is (0, 1, 0).

Direction cosines of the z-axis is (0, 0, 1).

9. Angle between two line segments:

If a₁, b₁, c₁ and a₂, b₂, c₂ are the direction ratios of two lines and θ is the acute angle between them, then

\begin{array}{l} c o s θ = | \frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{a_{1}^{2} + b_{1}^{2} + c_{1}^{2}} \sqrt{a_{2}^{2} + b_{2}^{2} + c_{2}^{2}}} | \end{array}

The line will be perpendicular if a₁a₂+ b₁b₂+ c₁c₂ = 0, and parallel if a₁/a₂ = b₁/b₂ = c₁/c₂.

10. Projection of a line segment on a line:

If P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) then the projection of PQ on a line having direction cosines l, m, n is |l(x₂– x₁) + m(y₂– y₁) + n(z₂– z₁)|

11. Equation of a plane: General form: ax + by + cz + d = 0, where a, b, c are not all zero, a, b, c, d ∈ R.

(i) Normal form: lx + my + nz = p

(ii) Plane through the point (x₁, y₁, z₁): a(x – x₁) + b(y – y₁) + c(z – z₁) = 0

(iii) Intercept form:

\begin{array}{l} \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \end{array}

(iv) vector form:

\begin{array}{l} (\vec{r} - \vec{a}) . \vec{n} = 0 \end{array}

\begin{array}{l} \vec{r} . \vec{n} = \vec{a} . \vec{n} \end{array}

(v) Planes parallel to the axes :

(a) plane parallel to X-axis is by + cz + d = 0

(b) plane parallel to Y-axis is ax + cz + d = 0

(vi) Plane through origin: Equation of the plane passing through the origin is ax + by + cz = 0.

(vii) Transformation of the equation of a plane to the normal form: ax+by+cz-d = 0 in normal form is

\begin{array}{l} \frac{a x}{\pm \sqrt{a^{2} + b^{2} + c^{2}}} + \frac{b y}{\pm \sqrt{a^{2} + b^{2} + c^{2}}} + \frac{c z}{\pm \sqrt{a^{2} + b^{2} + c^{2}}} = \frac{d}{\pm \sqrt{a^{2} + b^{2} + c^{2}}} \end{array}

(viii) Any plane parallel to the given plane ax + by + cz + d = 0 is ax + by + cz + λ = 0.

Distance between ax + by + cz + d₁ = 0 and ax + by + cz + d₂ = 0 is

\begin{array}{l} \frac{| d_{1} - d_{2} |}{\sqrt{a^{2} + b^{2} + c^{2}}} \end{array}

(ix) A plane ax + by + cz + d = 0 divides the line segment joining (x₁, y₁, z₁) and (x₂, y₂, z₂) in the ratio

\begin{array}{l} (- \frac{a x_{1} + b y_{1} + c z_{1} + d}{a x_{2} + b y_{2} + c z_{2} + d}) \end{array}

(x) Coplanarity of four points: The points A(x₁, y₁, z₁), B(x₂, y₂, z₂), C(x₃, y₃, z₃) and D(x₄, y₄, z₄) are coplanar if

\begin{array}{l} | \begin{array}{c} x_{2} - x_{1} & y_{2} - y_{1} & z_{2} - z_{1} \\ x_{3} - x_{1} & y_{3} - y_{1} & z_{3} - z_{1} \\ x_{4} - x_{1} & y_{4} - y_{1} & z_{4} - z_{1} \end{array} | = 0 \end{array}

12. A point and a plane:

(i) distance of the point (x’, y’, z’) from the plane ax+by+cz+d = 0 is given by

\begin{array}{l} \frac{a x^{‘} + b y^{‘} + c z^{‘} + d}{\sqrt{a^{2} + b^{2} + c^{2}}} \end{array}

(ii)

\begin{array}{l} Length of the perpendicular from a point \vec{a} to the plane \vec{r} . \vec{n} = d is given by p = \frac{| \vec{a} . \vec{n} - d |}{| \vec{n} |} \end{array}

(iii) Foot (x’, y’, z’) of perpendicular drawn from the point (x₁, y₁, z₁) to the plane ax+by+cz+d = 0 is given by

\begin{array}{l} \frac{x^{‘} - x_{1}}{a} = \frac{y^{‘} - y_{1}}{b} = \frac{z^{‘} - z_{1}}{c} = - \frac{a x_{1} + b y_{1} + c z_{1} + d}{a^{2} + b^{2} + c^{2}} \end{array}

(iv) To find image of a point with respect to a plane:

Let P (x₁, y₁, z₁) be a given point and ax + by + cz + d = 0 is given plane. Let (x’, y’, z’) is the image point. Then,

\begin{array}{l} \frac{x^{‘} - x_{1}}{a} = \frac{y^{‘} - y_{1}}{b} = \frac{z^{‘} - z_{1}}{c} = - 2 \frac{(a x_{1} + b y_{1} + c z_{1} + d)}{a^{2} + b^{2} + c^{2}} \end{array}

(v) The distance between two parallel planes ax + by + cz + d = 0 and ax + by + cz + d’ = 0 is given by

\begin{array}{l} \frac{| d - d^{'} |}{\sqrt{a^{2} + b^{2} + c^{2}}} \end{array}

13. Angle between two planes:

(i)

\begin{array}{l} \cos θ = | \frac{a a^{'} + b b^{'} + c c^{'}}{\sqrt{a^{2} + b^{2} + c^{2}} \sqrt{a^{' 2} + b^{' 2} + c^{' 2}}} | \end{array}

(ii) Planes are perpendicular if aa’ + bb’ + cc’ = 0 and parallel if a/a’ = b/b’ = c/c’.

(iii) The angle θ between the planes

\begin{array}{l} \vec{r} . \vec{n_{1}} = d_{1} \end{array}

and

\begin{array}{l} \vec{r} . \vec{n_{2}} = d_{2} \end{array}

is given by

\begin{array}{l} c o s θ = \frac{\vec{n_{1}} . \vec{n_{2}}}{| \vec{n_{1}} | . | \vec{n_{2}} |} \end{array}

(iv) Planes are perpendicular if

\begin{array}{l} \vec{n_{1}} . \vec{n_{2}} = 0 \end{array}

and planes are parallel if

\begin{array}{l} \vec{n_{1}} = λ \vec{n_{2}} \end{array}

, where λ is a scalar.

14. Angle bisectors:

(i) The equations of a planes bisecting the angle between two given planes a₁x + b₁y + c₁z + d₁ = 0 and a₂x + b₂y + c₂z + d₂ = 0 are

\begin{array}{l} \frac{a_{1} x + b_{1} y + c_{1} z + d_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2} + c_{1}^{2}}} = \pm \frac{a_{2} x + b_{2} y + c_{2} z + d_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2} + c_{2}^{2}}} \end{array}

(ii) Bisector of acute or obtuse angle: First, make both the constant terms positive. Then,

a₁a₂+ b₁b₂+ c₁c₂ > 0 ⇒ origin lies on obtuse angle.

a₁a₂+ b₁b₂+ c₁c₂ < 0 ⇒ origin lies on acute angle.

15. Area of a triangle:

Let A(x₁, y₁, z₁), B(x₂, y₂, z₂), C(x₃, y₃, z₃) be the vertices of a triangle, then

\begin{array}{l} Δ = \sqrt{Δ_{x}^{2} + Δ_{y}^{2} + Δ_{z}^{2}} \end{array}

where

\begin{array}{l} Δ_{x} = \frac{1}{2} | \begin{array}{c} y_{1} & z_{1} & 1 \\ y_{2} & z_{2} & 1 \\ y_{3} & z_{3} & 1 \end{array} | \end{array}

\begin{array}{l} Δ_{y} = \frac{1}{2} | \begin{array}{c} z_{1} & x_{1} & 1 \\ z_{2} & x_{2} & 1 \\ z_{3} & x_{3} & 1 \end{array} | \end{array}

and

\begin{array}{l} Δ_{z} = \frac{1}{2} | \begin{array}{c} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{array} | \end{array}

Vector method:

\begin{array}{l} From two vector \vec{A B} and \vec{A C} the area is given by \frac{1}{2} | \vec{A B} \times \vec{A C} | \end{array}

16. Volume of a tetrahedron:

Volume of a tetrahedron with vertices A(x₁, y₁, z₁), B(x₂, y₂, z₂), C(x₃, y₃, z₃) and D(x₄, y₄, z₄) is given by

\begin{array}{l} V = \frac{1}{6} | \begin{array}{c} x_{1} & y_{1} & z_{1} & 1 \\ x_{2} & y_{2} & z_{2} & 1 \\ x_{3} & y_{3} & z_{3} & 1 \\ x_{4} & y_{4} & z_{4} & 1 \end{array} | \end{array}

17. Equation of a line:

(i) A straight line is the intersection of two planes. It is represented by two planes a₁x + b₁y + c₁z + d₁ = 0 and a₂x + b₂y + c₂z + d₂ = 0 .

(ii) Symmetric form:

\begin{array}{l} \frac{(x - x_{1})}{a} = \frac{(y - y_{1})}{b} = \frac{(z - z_{1})}{c} = r \end{array}

(iii) vector equation:

\begin{array}{l} \vec{r} = \vec{a} + λ \vec{b} \end{array}

(iv) Reduction of cartesian form of equation of a line to vector form and vice versa

\begin{array}{l} \frac{x - x_{1}}{a} = \frac{y - y_{1}}{b} = \frac{z - z_{1}}{c} \end{array}

⇔

\begin{array}{l} \vec{r} = (x_{1} \hat{i} + y_{1} \hat{j} + z_{1} \hat{k}) + λ (a \hat{i} + b \hat{j} + c \hat{k}) \end{array}

18. To find image of a point with respect to a line:

\begin{array}{l} Let L = \frac{x - x_{2}}{a} = \frac{y - y_{2}}{b} = \frac{z - z_{2}}{c} be the given line. \end{array}

Let (x’, y’, z’) be the image of the point (x₁, y₁, z₁) with respect to the line L. Then

(i) a(x₁– x’) + b(y₁– y’) + c(z₁– z’) = 0

(ii)

\begin{array}{l} \frac{\frac{x_{1} + x^{'}}{2} - x_{2}}{a} = \frac{\frac{y_{1} + y^{'}}{2} - y_{2}}{b} = \frac{\frac{z_{1} + z^{'}}{2} - z_{2}}{c} = λ \end{array}

From (ii) get the value of x’, y’, z’ in terms of λ as x’= 2aλ + 2x₂– x₁, y’= 2bλ + 2y₂– y₁,

z’= 2cλ+2z₂-z₁ . Then put the values of x’, y’, z’ in (i) to get λ and substitute the value of λ to get (x’, y’,z’).

19. Angle between a line and a plane:

(i) If θ is the angle between a line

\begin{array}{l} \frac{x - x_{1}}{l} = \frac{y - y_{1}}{m} = \frac{z - z_{1}}{n} \end{array}

and the plane ax+by+cz+d = 0, then

\begin{array}{l} \sin θ = | \frac{a l + b m + c n}{\sqrt{a^{2} + b^{2} + c^{2}} \sqrt{l^{2} + m^{2} + n^{2}}} | \end{array}

(ii) Vector form : If θ is the angle between a line

\begin{array}{l} \vec{r} = \vec{a} + λ \vec{b} \end{array}

and

\begin{array}{l} \vec{r} . \vec{n} = d \end{array}

then

\begin{array}{l} s i n θ = \frac{\vec{b} . \vec{n}}{| \vec{b} | | \vec{n} |} \end{array}

(iii) condition for perpendicularity l/a = m/b = n/c ,

\begin{array}{l} \vec{b} \times \vec{n} = 0 \end{array}

(iv) condition for parallel : al + bm + cn = o,

\begin{array}{l} \vec{b} . \vec{n} = 0 \end{array}

20. Condition for a line to lie in a plane:

(i) cartesian form: Line

\begin{array}{l} \frac{x - x_{1}}{l} = \frac{y - y_{1}}{m} = \frac{z - z_{1}}{n} \end{array}

would lie in a plane ax+by+cz+d = 0, if ax₁+ by₁+ cz₁+ d = 0 and al + bm + cn = 0

(ii) Vector form:

\begin{array}{l} Line \vec{r} = \vec{a} + λ b would lie in the plane \vec{r} . \vec{n} = d if \vec{b} . \vec{n} = 0 and \vec{a} . \vec{n} = d \end{array}

21. Skew lines:

(i) The straight lines which are not parallel and non-coplanar are called skew lines.

\begin{array}{l} If Δ = | \begin{array}{c} α^{‘} - α & β^{'} - β & γ^{'} - γ \\ l & m & n \\ l^{'} & m^{'} & n^{'} \end{array} | \neq 0, then the lines are skew. \end{array}

(ii) Shortest distance:

\begin{array}{l} S D = \frac{| \begin{array}{c} α^{‘} - α & β^{'} - β & γ^{'} - γ \\ l & m & n \\ l^{'} & m^{'} & n^{'} \end{array} |}{\sqrt{\sum (m n^{'} - m^{'} n)^{2}}} \end{array}

(iii) Vector form:

\begin{array}{l} For lines \vec{r} = \vec{a_{1}} + λ \vec{b_{1}} and \vec{r} = \vec{a_{2}} + λ \vec{b_{2}} to be skew (\vec{b_{1}} \times \vec{b_{2}}) . (\vec{a_{2}} - \vec{a_{1}}) \neq 0 \end{array}

(iv) Shortest distance between lines

\begin{array}{l} \vec{r} = \vec{a} + λ \vec{b} \end{array}

and

\begin{array}{l} \vec{r} = \vec{a_{2}} + μ \vec{b} \end{array}

\begin{array}{l} d = | \frac{\vec{b} \times (\vec{a_{2}} - {\vec{a}}_{1})}{| \vec{b} |} | \end{array}

22. Family of planes:

(i) Any plane through the intersection of a₁x + b₁y + c₁z + d₁ = 0 and a₂x + b₂y + c₂z + d₂ = 0 is a₁x + b₁y + c₁z + d₁+ λ(a₂x + b₂y + c₂z + d₂) = 0

(ii) The equation of plane passing through the intersection of the planes

\begin{array}{l} \vec{r} . \vec{n_{1}} = d_{1} \end{array}

and

\begin{array}{l} \vec{r} . \vec{n_{2}} = d_{2} \end{array}

\begin{array}{l} \vec{r} . (\vec{n_{1}} + λ \vec{n_{2}}) = d_{1} + λ d_{2} \end{array}

where λ is an arbitrary scalar.

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