By going through these CBSE Class 12 Maths Notes Chapter 11 Three Dimensional Geometry, students can recall all the concepts quickly.

## Three Dimensional Geometry Notes Class 12 Maths Chapter 11

**1. Direction cosines of a line:**

Let AB be a line in space.

Through O, draw a line OP parallel to AB. Let OP makes angles α, β, and γ with OX, OY, and OZ respectively.

Cosines of the angles α, β, and γ,

i.e., cos α, cos β, and cos γ are known as the direction cosines of line AB.

Let l = cos α, m = cos β, n = cos γ

⇒ l, m, n are the direction cosines of the line AB Let us consider the ray BA. OQ is drawn parallel to BA. Now, OQ makes angles n – α, n – β, and n – γ with coordinates axes OX, OY, and OZ respectively.

∴ Direction cosines of BA are cos(π – α), cos (π – β) and cos (π – γ),

i.e., – cos α, – cos β, – cos γ or -l, -m, -n.

→ Relation between l, m, and n

AB is any ray having direction cosines l, m, n. Now OP is drawn parallel to AB, where P is the point (x, y, z). Let PM be drawn perpendicular to OY.

In ΔOPM OP = r (say). PM⊥OY.

∴ ∠PMO = 90°

Also, ∠POM = β.

∴ \(\frac{\mathrm{OM}}{\mathrm{OP}}\) = β

∴ \(\frac{y}{r}\) = m,

∴ y = rm

Similarly, x = rl and z = rn.

Now, OP^{2} = x^{2} + y^{2} + z^{2}

r^{2} = (rl)^{2} + (rm)^{2} + (rn)^{2
}or

r^{2} = r^{2}(l^{2} + m^{2} + n^{2})

⇒ l^{2} + m^{2} + n^{2} = 1.

→ Direction Ratios of a line

Definition: The numbers which are proportional to the direction cosines of a line are known as direction ratios of the line.

Let Z, m, n be the direction cosines of a line.

Multiplying each by r, we ger rl, rm, rn the direction ratios.

Let rl = a,rm = b and rn – c.

Squaring and adding

r^{2}l^{2} + r^{2}m^{2} + r^{2}n^{2} = a^{2} + b^{2} + c^{2}

∴ r^{2}(l^{2} + m^{2} + n^{2}) = a^{2} + b^{2} + c^{2}

∵ l^{2} + m^{2} + n^{2} = 1

∴ l = \(\frac{a}{r}=\frac{a}{\sqrt{a^{2}+b^{2}+c^{2}}}\)

Similarly, m = \(\frac{b}{\sqrt{a^{2}+b^{2}+c^{2}}}\) and n = \(\frac{c}{\sqrt{a^{2}+b^{2}+c^{2}}}\)

Thus, if a, b and c are the direction ratios of a line, the direction cosines are \(\frac{a}{\sqrt{a^{2}+b^{2}+c^{2}}}\), \(\frac{b}{\sqrt{a^{2}+b^{2}+c^{2}}}\) and \(\frac{c}{\sqrt{a^{2}+b^{2}+c^{2}}}\).

→ Direction cosines of the line passing through the points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2})

Direction ratios of the line PQ are x_{2} – x_{1}, y_{2} – y_{1,} z_{2} – z_{1}

∴ PQ = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}\)

∴Direction cosines of PQ are \(\frac{x_{2}-x_{1}}{\mathrm{PQ}}\), \(\frac{y_{2}-y_{1}}{\mathrm{PQ}}\), \(\frac{z_{2}-z_{1}}{P Q}\)

**2. Angle between the two lines**

1. Let l_{1,} m_{1}, n_{1}, and l_{2}, m_{2}, n_{2} be the direction cosines of the lines OP and OQ.

The angle θ between these lines is given by

cos θ = l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2}

and sin θ = \(\sqrt{\left(m_{1} n_{2}-m_{2} n_{1}\right)^{2}+\left(n_{1} l_{2}-n_{2} l_{1}\right)^{2}+\left(l_{1} m_{2}-l_{2} m_{1}\right)^{2}}\)

2. If a_{1}, b_{1}, c_{1} and a_{2}, b_{2}, c_{2} are the direction ratios of two lines, then

3. Two lines are perpendicular to each other,

if θ = \(\frac{π}{2}\) ⇒ cos θ = cos \(\frac{π}{2}\) = 0

⇒ l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2} = 0

or

a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0

4. When the lines are parallel

θ = 0 ⇒ sin θ = sin 0 = 0.

**STRAIGHT LINE**

**3. Equation of a line through a given point:**

(a) Let the line passes through \(\vec{a}\) and is parallel to vector \(\vec{b}\)

Then, the equation of the line is

\(\vec{r}\) – \(\vec{a}\) = λ\(\vec{b}\)

or

\(\vec{r}\) = \(\vec{a}\) + λ\(\vec{b}\)

(b) Let the point A be (x_{1}, y_{1}, z_{1}) and a, b, c are the direction ratios of the line. The equation of the line is

\(\frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}=\frac{z-z_{1}}{c}\)

If l, m, n are the direction cosines, then equation of the line is

\(\frac{x-x_{1}}{l}=\frac{y-y_{1}}{m}=\frac{z-z_{1}}{n}\)

**4. Equation of the line passing through two points:**

(a) Let \(\vec{a}_{1}\), \(\vec{a}_{2}\) be the position vectors of two points P and Q respectively.

⇒ \(\vec{b}\) = \(\vec{a}_{1}\) – \(\vec{a}_{2}\)

∴ Equation of PQ is

\(\vec{r}\) = \(\vec{a}\) + λ(\(\vec{a}_{2}\) – \(\vec{a}_{1}\))

(b) Direction ratios of the line passing through P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) are x_{2} – x_{1}, y_{2} – y_{1} and z_{2} – z_{1}

∴ Equation of the line PQ is

\(\frac{x-x_{1}}{x_{2}-x_{1}}=\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{z-z_{1}}{z_{2}-z_{1}}\)

**5. Angle between two straight lines:**

(a) Let the two lines be

\(\vec{r}\) = \(\vec{a}_{1}\) + λ\(\vec{b}_{1}\)

\(\vec{r}\) = \(\vec{a}_{2}\) + λ\(\vec{b}_{2}\)

If θ be the angle between them, then

cos θ = \(\frac{\vec{b}_{1} \vec{b}_{2}}{\left|\vec{b}_{1}\right|\left|\vec{b}_{2}\right|}\)

(b) Let the lines be

\(\frac{x-x_{1}}{a_{1}}=\frac{y-y_{1}}{b_{1}}=\frac{z-z_{1}}{c_{1}}\)

and \(\frac{x-x_{2}}{a_{2}}=\frac{y-y_{2}}{b_{2}}=\frac{z-z_{2}}{c_{2}}\)

i.e., a1, b1, c1 and a2, b2, c2 are the direction ratios of the lines.

∴ cos θ = \(\frac{a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\)

If l1, m1, n1 and l_{2}, m_{2}, n_{2} are the direction cosines, then

cos θ = l_{1}l_{2} + m1m_{2} +n1n_{2}

**6. Shortest Distance:**

(a)Let \(\vec{r}\) = \(\vec{a}_{1}\) + λ\(\vec{b}_{1}\) and

\(\vec{r}\) = \(\vec{a}_{2}\) + λ\(\vec{b}_{2}\) be the two non-intersecting lines.

The shortest distance between the given lines = \(\left|\frac{\left(\vec{b}_{1} \times \vec{b}_{2}\right) \cdot\left(\vec{a}_{2}-\vec{a}_{1}\right)}{\left|\vec{b}_{1} \times \vec{b}_{2}\right|}\right|\)

(b) Let the lines be

If the lines are intersecting, then lines are coplanar.

⇒ S.D. = 0

⇒ (\(\vec{b}_{1} \times \vec{b}_{2}\)) – (\(\vec{a}_{1} \times \vec{a}_{2}\)) = 0

or

\(\left|\begin{array}{ccc}

x_{2}-x_{1} & y_{2}-y_{1} & z_{2}-z_{1} \\

a_{1} & b_{1} & c_{1} \\

a_{2} & b_{2} & c_{2}

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

**7. Distance between parallel lines:**

Let the parallel lines be

\(\vec{r}\) = \(\vec{a}_{1}\) + λ\(\vec{b}_{1}\) and

\(\vec{r}\) = \(\vec{a}_{2}\) + μ\(\vec{b}_{2}\)

Shortest distance d between these lines is given by,

d = \(\left|\frac{\vec{b} \times\left(\vec{a}_{1}-\vec{a}_{2}\right)}{|\vec{b}|}\right|\)

**PLANES**

**8. Different forms of equations of a plane:**

1. Normal Form.

(a) Let the plane ABC be at a distance d from the origin. ON is the normal to the plane in the directon n̂. Equation of the plane is \(\vec{r}\). n̂ = d.

(b) If l, m, n are the direction cosines of the normal to the plane which is at distance d from the origin. The equation of the plane is lx + my + nz = d.

However, general form of the equation of a plane are \(\vec{r}\) .\(\overrightarrow{\mathrm{N}}\) = D and Ax + By + Cz + D = 0.

2. (a) Let the plane passes through a point A and let it perpendicular to the vector \(\overrightarrow{\mathrm{N}}\)

∴ Equation of the plane

(\(\vec{r}\) – \(\vec{a}\)).\(\overrightarrow{\mathrm{N}}\) = 0.

(b) If a plane passes through (x_{1}, y_{1}, z_{1}) and perpendicular to the line with direction ratios a, b, c the equation of the plane is

a(x – x_{1}) + b(y – y_{1})+c(z – z_{1}) =0

3. Equation of the plane passing through three points.

(a) Let the three points be A, B and C whose position vectors be, \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\)

The equation of the plane is

\((\vec{r}-\vec{a}) \cdot[(\vec{b}-\vec{a}) \times(\vec{c}-\vec{a})]\) = 0

(b) Let the three points through which the plane is passing be A(x_{1}, y_{1}, z_{1}), B(x_{2}, y_{2}, z_{2}), and C(x_{3}, y_{3}, z_{3}).

Then, equation of the plane is

\(\left|\begin{array}{llll}

x & y & z & 1 \\

x_{1} & y_{1} & z_{1} & 1 \\

x_{2} & y_{2} & z_{2} & 1 \\

x_{3} & y_{3} & z_{3} & 1

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

or

\(\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

4. Intercepts form of the equation of the plane

Let the plane make the intercepts a, b, and c on coordinate axes OX, OY, and OZ respectively.

Then, the equation of the plane is

\(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\) = 1

**9. Plane passing through the intersection of the two planes:**

Let the equation of two planes be

\(\vec{r}\). \(\vec{n}_{1}\) = d1 and

\(\vec{r}\) .\(\vec{n}_{2}\) = d2

The equation of the plane passing through the line of intersection of the given planes is

(\(\vec{r}\). \(\vec{n}_{1}\) – d1) + λ(\(\vec{r}\) .\(\vec{n}_{2}\) – d2) = 0

or

\(\vec{r}\) .(\(\vec{n}_{1}\) + λ\(\vec{n}_{2}\)) = (d1 + λd2)

(b) Equation of the plane passing through the line of intersection of 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 (a_{1}x + b_{1} y + c_{1}z + d_{1} + λ(a_{2}x + b_{2}y + c_{2}z + d_{2}) = 0

or

(a_{1} + λa_{2})x + (b_{1} + λb_{2})y + (c_{1} + λc_{2})z + d_{1} + λd_{2} = 0,

where λ is determined according to the given condition.

**10. Coplanarity of two lines:**

(a) The two lines

\(\vec{r}\) = \(\vec{a}_{1}\) + λ\(\vec{b}_{1}\) and

\(\vec{r}\) = \(\vec{a}_{2}\) + λ\(\vec{b}_{2}\) intersect each other, if \(\left(\vec{a}_{2}-\vec{a}_{1}\right) \cdot\left(\vec{b}_{2} \times \vec{b}_{1}\right)\) = 0

**11. The angle between the planes:**

Definition : The angle between the two planes is the angle between their normals. The angle 0 between the planes

\(\vec{r}\) = \(\vec{n}_{1}\) = d1 and \(\vec{r}\) = \(\vec{n}_{1}\) = d2 is given by cos θ = \(\frac{\left|\vec{n}_{1} \cdot \vec{n}_{2}\right|}{\left|\vec{n}_{1}\right|\left|\vec{n}_{2}\right|}\)

(b) If the planes are

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 the angle θ between these planes is given by

cos θ = \(\frac{a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\)

(c) The planes are perpendicular to each other, if θ = \(\frac{π}{2}\)

∴ cos \(\frac{π}{2}\) = 0 ⇒ \(\vec{n}_{1}\).\(\vec{n}_{2}\) = 0

or

a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0

(d) 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} = O

are parallel, if \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\).

(e) Equation of the plane parallel to

\(\vec{r} \cdot \vec{n}\) = d is \(\vec{r} \cdot \vec{n}\) = λ

Plane parallel to

ax + by + cz + d = 0 is ax + by + cz + λ = 0.

**12. Distance of a point from th plane:**

(a) Let the plane be \(\vec{r} \cdot \vec{n}\) = d.

∴ Perpendicular distance of the point \(\vec{a}\) from the plane

=|d – \(\vec{a} \cdot \hat{n}\)|

(b) Let the equation of the plane

be Ax + By + Cz + D= O.

The distance of the point (x1, y1, z1) from this plane

= \(\left|\frac{A x_{1}+B y_{1}+C z_{1}+D}{\sqrt{A^{2}+B^{2}+C^{2}}}\right|\)

**13. Angle between a line and a plane:**

Definition: The angle between a line and plane is said to be the complement of the angle between the line and the normal to the plane.

(a) Let the line and plane be

\(\vec{r}\) = \(\vec{a}\) + λ\(\vec{b}\) and \(\vec{r} \cdot \vec{n}\) = d.

If θ be the angle between the plane and the line, then

sin θ = \(\frac{\vec{b} \cdot \vec{n}}{|\vec{b}||\vec{n}|}\)

(b) ¡et the line and the plane be \(\frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}=\frac{z-z_{1}}{c}\) and Ax + By + Cz + D = O.

The angle Φ between them is given by

sin Φ = \(\frac{a \cdot \mathrm{A}+b \cdot \mathrm{B}+c \cdot \mathrm{C}}{\sqrt{a^{2}+b^{2}+c^{2}} \sqrt{\mathrm{A}^{2}+\mathrm{B}^{2}+\mathrm{C}^{2}}}\).

1. IMPORTANT RESULT

l^{2} + m^{2} + n^{2} = 1, where < l, m, n > are direction-cosines of a st. line.

2. DIRECTION-RATIOS

The direction-ratios of the line joining of the points (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) are :

<x_{2}-x_{1}, y_{2}-y_{1} ,z_{2}-z_{1} >.

3. (i) Angle between two lines. The angle between two lines having direction-cosines

< l_{1} m_{1} n_{1} > and < l_{2}, m_{2}, n_{2} > is given by :

cos θ = |l_{1}l_{2}+ m_{1}m_{2} + n_{1}n_{2}|.

(ii) The lines are:

(a) perpendicular l_{1}l_{2}+ m_{1}m_{2} + n_{1}n_{2 }= 0.

(b) parallel iff l_{1 }= l_{2, }m_{1 }= m_{2}, + n_{1 }= n_{2 }

4. SHORTEST DISTANCE

The shortest distance between two lines :

\(\vec{r}=\overrightarrow{a_{1}}+\lambda \overrightarrow{b_{1}} \text { and } \vec{r}=\overrightarrow{a_{2}}+\mu \overrightarrow{b_{2}} \text { is }\left|\frac{\left(\overrightarrow{b_{1}} \times \overrightarrow{b_{2}}\right) \cdot\left(\overrightarrow{a_{2}}-\overrightarrow{a_{1}}\right)}{\left|\overrightarrow{b_{1}} \times \overrightarrow{b_{2}}\right|}\right|\)

5. EQUATIONS OF PLANES

(i) Equation of a plane, which is at a distance ‘p’ from the origin and perpendicular to the unit

vector \(\hat{n}\) is \(\vec{r} \cdot \hat{n}=p\)

(ii) General Form. The general equation of first degree i.e.ax + by + cz + d- 0 represents a plane,

(iii) One-point Form. The equation of a plane through (*,, y,, z,) and having <a,b,c> as direction-

ratios of the normal is a (x – x_{1}) + b (y – y_{1}) + c (z – z_{1}) = 0.

(iv) Three-point Form. The equation of the plane through (x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}) and (x_{3}, y_{3}, z_{3}) is:

\(\left|\begin{array}{ccc}

x-x_{1} & y-y_{1} & z-z_{1} \\

x-x_{2} & y-y_{2} & z-z_{2} \\

x-x_{3} & y-y_{3} & z-z_{3}

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

(v) Intercept Form. Equation of the plane, which cuts off intercepts a, b, c on the axes, is :

\(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\) = 1

6. ANGLE BETWEEN TW O PLANES

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 given by :

cos θ = \(\frac{\left|a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}\right|}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\)

7. BISECTING PLANES

The equations of the planes bisecting 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 are:

\(\frac{a_{1} x+b_{1} y+c_{1} z+d_{1}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}}=\pm \frac{a_{2} x+b_{2} y+c_{2} z+d_{2}}{\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\)