Hello Students. In this post, you will find the complete** ****NCERT Solutions for Maths Class 12 Exercise 11.2**.

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**NCERT Solutions for Maths Class 12 Exercise 11.1**

**NCERT Solutions for Maths Class 12 Exercise 11.2**

**Maths Class
12 Ex 11.2 Question 1.**

Show that the three lines with direction cosines
(12/13, –3/13, –4/13); (4/13, 12/13, 3/13); (3/13, –4/13,
12/13); are mutually perpendicular.

**Solution:**

Let the lines be L_{1},
L_{2} and L_{3}.

∴ For line L_{1}, direction
cosines are: *l*_{1} = 12/13;
*m*_{1} = –3/13;
*n*_{1} = –4/13;

For line L_{2}, direction cosines are: *l*_{2} = 4/13; *m*_{2}
= 12/13; *n*_{2}
= 3/13;

For line L_{3}, direction cosines are: *l*_{3} = 3/13; *m*_{3}
= –4/13; *n*_{3}
= 12/13;

Let the angle between lines L_{1 }and L_{2 }is Î¸_{1}.

Then, cos Î¸_{1
}= *l*_{1}* l*_{2 }+ *m*_{1}* m*_{2 }+
*n*_{1}* n*_{2}

= (12/13 × 4/13) + (–3/13
× 12/13) + (–4/13 × 3/13)

= 48/169 – 36/169 –
12/169 = 0

∴
Î¸_{1} = 90˚

Thus,
the angle between lines L_{1
}and L_{2 }is 90˚.

Let the angle between lines L_{2 }and L_{3 }is Î¸_{2}.

Then, cos Î¸_{2
}= *l*_{2}* l*_{3 }+ *m*_{2}* m*_{3 }+
*n*_{2}* n*_{3}

= (4/13 × 3/13) + (12/13 × –4/13)
+ (3/13 × 12/13)

= 12/169 – 48/169 + 36/169
= 0

∴
Î¸_{2} = 90˚

Thus,
the angle between lines L_{2
}and L_{3 }is 90˚.

Let the angle between lines L_{1 }and L_{3 }is Î¸_{3}.

Then, cos Î¸_{3
}= *l*_{1}* l*_{3 }+ *m*_{1}* m*_{3 }+
*n*_{1}* n*_{3}

= (12/13 × 3/13) + (–3/13
× –4/13) + (–4/13 × 12/13)

= 36/169 + 12/169 – 48/169
= 0

∴
Î¸_{3} = 90˚

Thus,
the angle between lines L_{1
}and L_{3 }is 90˚.

Hence,
the given lines are mutually perpendicular.

**Maths Class
12 Ex 11.2 Question 2.**

Show that the
line through the points (1, –1, 2), (3, 4, –2) is perpendicular to the line
through the points (0, 3, 2) and (3, 5, 6).

**Solution:**

Let A and B be
the points (1, –1, 2) and (3, 4, –2), respectively.

Direction ratios
of AB are: a_{1} = 3 – 1 = 2, b_{1} = 4 – (–1) = 5, c_{1}
= –2 – 2 = –4

Let C and D be
the points (0, 3, 2) and (3, 5, 6), respectively.

Direction ratios
of CD are: a_{2} = 3 – 0 = 3, b_{2} = 5 – 3 = 2, c_{2}
= 6 – 2 = 4

Now, AB is perpendicular
to CD if

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

2 × 3 + 5 × 2 +
(–4) × 4 = 6 + 10 – 16 = 0

Hence, the lines
AB and CD are perpendicular.

**Maths Class
12 Ex 11.2 Question 3.**

Show that the
line through the points (4, 7, 8), (2, 3, 4) is parallel to the line through the
points (–1, –2, 1) and (1, 2, 5).

**Solution:**

Let the points be
A(4, 7, 8), B(2, 3, 4), C(–1, –2, 1) and D(1, 2, 5).

Now, direction ratios of AB are:

**Maths Class
12 Ex 11.2 Question 4.**

Find the equation
of the line which passes through the point (1, 2, 3) and is parallel to the
vector 3** i** + 2

**– 2**

*j***.**

*k***Solution:**

Equation of the
line passing through the point

**Maths Class
12 Ex 11.2 Question 5.**

Find the equation
of the line in vector and in Cartesian form that passes through the point with
position vector 2** i** –

**+ 4**

*j***and is in the direction**

*k***+ 2**

*i***–**

*j***.**

*k***Solution:**

The vector
equation of a line passing through a point with position vector ** a** = 2

**–**

*i***+ 4**

*j***and parallel to the vector**

*k***=**

*b***+ 2**

*i***–**

*j***is**

*k***Maths Class
12 Ex 11.2 Question 6.**

Find the Cartesian equation of the line which passes through the point P(–2, 4, –5) and parallel to the line is given by.

**Solution:**

Given that the line passes through the point (–2, 4, –5) and parallel to the line given by .

Thus, the line
passing through P(x_{1}, y_{1}, z_{1}) where x_{1}
= –2, y_{1} = 4, z_{1} = –5 and the direction ratios a_{1}
= 3, b_{1} = 5, c_{1} = 6 is

**Maths Class
12 Ex 11.2 Question 7.**

The Cartesian equation of a line is. Write its vector form.

**Solution:**

The Cartesian equation of the line is … (i)

Clearly (i) passes through the point (5, –4, 6) and has 3, 7, 2 as its
direction ratios.

Line (i) passes through the point A with

**Maths Class
12 Ex 11.2 Question 8.**

Find the angle
between the following pairs of lines

**Solution:**

**(i)** Let Î¸ be the angle between the given lines.

The given lines are parallel to the vectors

**Maths Class
12 Ex 11.2 Question 9.**

Find the angle
between the following pairs of lines

**Solution:**

**Maths Class
12 Ex 11.2 Question 10.**

Find the values
of *p* so that the lines

are at right angles.

**Solution:**

The given
equations are not in the standard form.

The equations of given lines in standard form are

**Maths Class
12 Ex 11.2 Question 11.**

Show that the linesare perpendicular to each other

**Solution:**

Then the direction ratios of the given lines are: *a*_{1} = 7, *b*_{1}
= –5, *c*_{1} = 1 and *a*_{2} = 1, *b*_{2} = 2, *c*_{2}
= 3

**Maths Class
12 Ex 11.2 Question 12.**

Find the shortest
distance between the lines

**Solution:**

**Maths Class
12 Ex 11.2 Question 13.**

Find the shortest
distance between the lines

**Solution:**

Shortest distance
between the lines

**Maths Class
12 Ex 11.2 Question 14.**

Find the shortest
distance between the lines whose vector equations are:

**Solution:**

**Maths Class
12 Ex 11.2 Question 15.**

Find the shortest
distance between the lines whose vector equations are

**Solution:**

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