NCERT Solutions for Maths Class 12 Exercise 11.2
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Class 12th is a very crucial stage of your student’s life, since you take all important decisions about your career on this stage. Mathematics plays a vital role to take decision for your career because if you are good in mathematics, you can choose engineering and technology field as your career.
NCERT Solutions for Maths Class 12 Exercise 11.2 helps you to solve each and every problem with step by step explanation which makes you strong in mathematics.
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NCERT Solutions for Maths Class 12 Exercise 11.1
NCERT Solutions for Maths Class 12 Exercise 11.3
NCERT Solutions for Maths Class 12 Exercise 11.2
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 L1,
L2 and L3.
∴ For line L1, direction
cosines are: l1 = 12/13;
m1 = –3/13;
n1 = –4/13;
For line L2, direction cosines are: l2 = 4/13; m2
= 12/13; n2
= 3/13;
For line L3, direction cosines are: l3 = 3/13; m3
= –4/13; n3
= 12/13;
Let the angle between lines L1 and L2 is θ1.
Then, cos θ1
= l1 l2 + m1 m2 +
n1 n2
= (12/13 × 4/13) + (–3/13
× 12/13) + (–4/13 × 3/13)
= 48/169 – 36/169 –
12/169 = 0
∴
θ1 = 90˚
Thus,
the angle between lines L1
and L2 is 90˚.
Let the angle between lines L2 and L3 is θ2.
Then, cos θ2
= l2 l3 + m2 m3 +
n2 n3
= (4/13 × 3/13) + (12/13 × –4/13)
+ (3/13 × 12/13)
= 12/169 – 48/169 + 36/169
= 0
∴
θ2 = 90˚
Thus,
the angle between lines L2
and L3 is 90˚.
Let the angle between lines L1 and L3 is θ3.
Then, cos θ3
= l1 l3 + m1 m3 +
n1 n3
= (12/13 × 3/13) + (–3/13
× –4/13) + (–4/13 × 12/13)
= 36/169 + 12/169 – 48/169
= 0
∴
θ3 = 90˚
Thus,
the angle between lines L1
and L3 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: a1 = 3 – 1 = 2, b1 = 4 – (–1) = 5, c1
= –2 – 2 = –4
Let C and D be
the points (0, 3, 2) and (3, 5, 6), respectively.
Direction ratios
of CD are: a2 = 3 – 0 = 3, b2 = 5 – 3 = 2, c2
= 6 – 2 = 4
Now, AB is perpendicular
to CD if
a1a2
+ b1b2 + c1c2 = 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 3i + 2j – 2k.
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 2i – j + 4k and is in the direction i
+ 2j
– k.
Solution:
The vector
equation of a line passing through a point with position vector a = 2i
– j
+ 4k and
parallel to the vector b = i + 2j – k is
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(x1, y1, z1) where x1
= –2, y1 = 4, z1 = –5 and the direction ratios a1
= 3, b1 = 5, c1 = 6 is
Maths Class
12 Ex 11.2 Question 7.
The Cartesian
equation of a line is
Solution:
The Cartesian
equation of the line is
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
Find the vector
and the Cartesian equations of the lines that passes through the origin and (5,
–2, 3).
Solution:
Given that the
line passes through point A (0, 0, 0) and B(5, –2, 3).
Therefore, a = 0i + 0j + 0k and b
= 5i
– 2j
+ 3k
Direction ratios of the line passing through the
Maths Class
12 Ex 11.2 Question 9.
Find the vector
and Cartesian equations of the line that passes through the points (3, –2, –5),
(3, –2, 6).
Solution:
The PQ passes
through the point P(3, –2, –5).
Maths Class
12 Ex 11.2 Question 10.
Find the angle
between the following pairs of lines
Solution:
(i) Let θ be the angle between the given lines.
The given lines are parallel to the vectors
Maths Class
12 Ex 11.2 Question 11.
Find the angle
between the following pairs of lines
Maths Class
12 Ex 11.2 Question 12.
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 13.
Show that the
lines
Solution:
The given lines
are
Then the direction ratios of the given lines are: a1 = 7, b1
= –5, c1 = 1 and a2 = 1, b2 = 2, c2
= 3
Maths Class
12 Ex 11.2 Question 14.
Find the shortest
distance between the lines
Solution:
Maths Class 12 Ex 11.2 Question 15.
Find the shortest
distance between the lines
Solution:
Shortest distance
between the lines
Maths Class
12 Ex 11.2 Question 16.
Find the shortest
distance between the lines whose vector equations are:
Solution:
Maths Class 12 Ex 11.2 Question 17.
Find the shortest
distance between the lines whose vector equations are
Solution:
