Chapter 11 Three Dimensional Geometry Ex 11.2 Solutions
Question - 1 : - Show that the three lines with direction cosines
Are mutually perpendicular.
Answer - 1 : -
Let us consider thedirection cosines of L1, L2 and L3 bel1, m1, n1; l2, m2, n2 andl3, m3, n3.
We know that
If l1, m1,n1 and l2, m2, n2 are thedirection cosines of two lines;
And θ is the acuteangle between the two lines;
Then cos θ = |l1l2 +m1m2 + n1n2|
If two lines areperpendicular, then the angle between the two is θ = 90°
For perpendicularlines, | l1l2 + m1m2 + n1n2 |= cos 90° = 0, i.e. | l1l2 + m1m2 +n1n2 | = 0
So, in order to checkif the three lines are mutually perpendicular, we compute | l1l2 +m1m2 + n1n2 | for all thepairs of the three lines.
Firstly let uscompute, | l1l2 + m1m2 +n1n2 |
So, L1⊥ L2 …… (1)
Similarly,
Let us compute, | l2l3 +m2m3 + n2n3 |
So, L2⊥ L3 ….. (2)
Similarly,
Let us compute, | l3l1 +m3m1 + n3n1 |
So, L1⊥ L3 ….. (3)
∴ By (1), (2) and(3), the lines are perpendicular.
L1, L2 andL3 are mutually perpendicular.
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).
Answer - 2 : -
Given:
The points (1, –1, 2),(3, 4, –2) and (0, 3, 2), (3, 5, 6).
Let us consider AB bethe line joining the points, (1, -1, 2) and (3, 4, -2), and CD be the linethrough the points (0, 3, 2) and (3, 5, 6).
Now,
The direction ratios,a1, b1, c1 of AB are
(3 – 1), (4 – (-1)),(-2 – 2) = 2, 5, -4.
Similarly,
The direction ratios,a2, b2, c2 of CD are
(3 – 0), (5 – 3), (6 –2) = 3, 2, 4.
Then, AB and CD willbe perpendicular to each other, if a1a2 + b1b2 +c1c2 = 0
a1a2 +b1b2 + c1c2 = 2(3) + 5(2)+ 4(-4)
= 6 + 10 – 16
= 0
∴ AB and CD areperpendicular to each other.
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), (1, 2, 5).
Answer - 3 : -
Given:
The points (4, 7, 8),(2, 3, 4) and (–1, –2, 1), (1, 2, 5).
Let us consider AB bethe line joining the points, (4, 7, 8), (2, 3, 4) and CD be the line through thepoints (–1, –2, 1), (1, 2, 5).
Now,
The direction ratios,a1, b1, c1 of AB are
(2 – 4), (3 – 7), (4 –8) = -2, -4, -4.
The direction ratios,a2, b2, c2 of CD are
(1 – (-1)), (2 –(-2)), (5 – 1) = 2, 4, 4.
Then AB will beparallel to CD, if
So, a1/a2 =-2/2 = -1
b1/b2 =-4/4 = -1
c1/c2 =-4/4 = -1
∴ We can say that,
-1 = -1 = -1
Hence, AB is parallelto CD where the line through the points (4, 7, 8), (2, 3, 4) is parallel to theline through the points (–1, –2, 1), (1, 2, 5)
Question - 4 : - Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector 
.
Answer - 4 : -
Question - 5 : - Find the equation of the line in vector and in Cartesian form that passes through the point with position vector 
and 
is in the direction
Answer - 5 : -
Question - 6 : - Find the Cartesian equation of the line which passes through the point (–2, 4, –5) and parallel to the line given by
Answer - 6 : -
Given:
The points (-2, 4, -5)
We know that
The Cartesian equationof a line through a point (x1, y1, z1) andhaving direction ratios a, b, c is
Question - 7 : - The Cartesian equation of a line is
. Write its vector form.
Answer - 7 : -
So when comparing thisstandard form with the given equation, we get
x1 =5, y1 = -4, z1 = 6 and
l = 3, m = 7, n = 2
Answer - 8 : -
Question - 9 : - Find the vector and the Cartesian equations of the line that passes through the points (3, –2, –5), (3, –2, 6).
Answer - 9 : -
Question - 10 : - Find the angle between the following pairs of lines:
Answer - 10 : -