NCERT Solutions for Class 12 Maths Chapter 11 – Three Dimensional Geometry Ex 11.2
Page No 477:
Question 1:
Show that the three lines with direction cosines
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7161/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_38ee7548.gif)
Answer:
Two lines with direction cosines, l1, m1, n1 and l2, m2, n2, are perpendicular to each other, if l1l2 + m1m2 + n1n2 = 0
(i) For the lines with direction cosines,
and
, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7161/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m2e430a61.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7161/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m5bf0233d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7161/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_4861155f.gif)
Therefore, the lines are perpendicular.
(ii) For the lines with direction cosines,
and
, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7161/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m5bf0233d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7161/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_md306b52.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7161/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m479a57d1.gif)
Therefore, the lines are perpendicular.
(iii) For the lines with direction cosines,
and
, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7161/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_md306b52.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7161/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m2e430a61.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7161/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m58d04951.gif)
Therefore, the lines are perpendicular.
Thus, all the lines 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:
Let AB be the line joining the points, (1, −1, 2) and (3, 4, − 2), and CD be the line joining the points, (0, 3, 2) and (3, 5, 6).
The direction ratios, a1, b1, c1, of AB are (3 − 1), (4 − (−1)), and (−2 − 2) i.e., 2, 5, and −4.
The direction ratios, a2, b2, c2, of CD are (3 − 0), (5 − 3), and (6 −2) i.e., 3, 2, and 4.
AB and CD will be perpendicular to each other, if a1a2 + b1b2+ c1c2 = 0
a1a2 + b1b2+ c1c2 = 2 × 3 + 5 × 2 + (− 4) × 4
= 6 + 10 − 16
= 0
Therefore, AB and CD are perpendicular 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:
Let AB be the line through the points, (4, 7, 8) and (2, 3, 4), and CD be the line through the points, (−1, −2, 1) and (1, 2, 5).
The directions ratios, a1, b1, c1, of AB are (2 − 4), (3 − 7), and (4 − 8) i.e., −2, −4, and −4.
The direction ratios, a2, b2, c2, of CD are (1 − (−1)), (2 − (−2)), and (5 − 1) i.e., 2, 4, and 4.
AB will be parallel to CD, if ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7164/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_5b8553c6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7164/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_5b8553c6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7164/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m6ec16d6d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7164/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_71045d43.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7164/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_6d7182d4.gif)
Thus, AB is parallel to CD.
Question 4:
Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7165/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_2d17951d.gif)
Answer:
It is given that the line passes through the point A (1, 2, 3). Therefore, the position vector through A is ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7165/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_31f2b942.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7165/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_31f2b942.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7165/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_33a2c750.gif)
It is known that the line which passes through point A and parallel to
is given by
is a constant.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7165/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_6d43734a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7165/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m2736d221.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7165/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m4cc56b35.gif)
This is the required equation of the line.
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
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7166/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_2caf1bf.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7166/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m6c6e4b8b.gif)
Answer:
It is given that the line passes through the point with position vector
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7166/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_54b2610.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7166/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m3b2eb094.gif)
It is known that a line through a point with position vector
and parallel to
is given by the equation, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7166/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m411cb30d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7166/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_4f2bfef5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7166/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_6d43734a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7166/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m411cb30d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7166/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_19c2617e.gif)
This is the required equation of the line in vector form.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7166/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_4bb281b7.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7166/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m780a9d4f.gif)
Eliminating λ, we obtain the Cartesian form equation as
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7166/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_9181804.gif)
This is the required equation of the given line in Cartesian form.
Question 6:
Find the Cartesian equation of the line which passes through the point
(−2, 4, −5) and parallel to the line given by![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7167/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m54251876.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7167/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m54251876.gif)
Answer:
It is given that the line passes through the point (−2, 4, −5) and is parallel to ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7167/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m54251876.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7167/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m54251876.gif)
The direction ratios of the line,
, are 3, 5, and 6.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7167/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m54251876.gif)
The required line is parallel to ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7167/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m54251876.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7167/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m54251876.gif)
Therefore, its direction ratios are 3k, 5k, and 6k, where k ≠ 0
It is known that the equation of the line through the point (x1, y1, z1) and with direction ratios, a, b, c, is given by ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7167/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_3f62e574.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7167/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_3f62e574.gif)
Therefore the equation of the required line is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7167/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_7cc8c4f5.gif)
Question 7:
The Cartesian equation of a line is
. Write its vector form.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7169/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_d8ad344.gif)
Answer:
The Cartesian equation of the line is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7169/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m6067b60d.gif)
The given line passes through the point (5, −4, 6). The position vector of this point is ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7169/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m51bc4a39.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7169/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m51bc4a39.gif)
Also, the direction ratios of the given line are 3, 7, and 2.
This means that the line is in the direction of vector, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7169/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m4e30e180.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7169/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m4e30e180.gif)
It is known that the line through position vector
and in the direction of the vector
is given by the equation, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7169/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m5a7e1003.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7169/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_4f2bfef5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7169/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_6d43734a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7169/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m5a7e1003.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7169/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m20838699.gif)
This is the required equation of the given line in vector form.
Question 8:
Find the vector and the Cartesian equations of the lines that pass through the origin and (5, −2, 3).
Answer:
The required line passes through the origin. Therefore, its position vector is given by,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7170/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m295dca29.gif)
The direction ratios of the line through origin and (5, −2, 3) are
(5 − 0) = 5, (−2 − 0) = −2, (3 − 0) = 3
The line is parallel to the vector given by the equation, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7170/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_55780ef8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7170/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_55780ef8.gif)
The equation of the line in vector form through a point with position vector
and parallel to
is, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7170/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_5277a5d3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7170/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_4f2bfef5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7170/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_6d43734a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7170/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_5277a5d3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7170/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_48674c35.gif)
The equation of the line through the point (x1, y1, z1) and direction ratios a, b, c is given by, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7170/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_3f62e574.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7170/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_3f62e574.gif)
Therefore, the equation of the required line in the Cartesian form is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7170/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m4fe9c54c.gif)
Page No 478:
Question 9:
Find the vector and the Cartesian equations of the line that passes through the points (3, −2, −5), (3, −2, 6).
Answer:
Let the line passing through the points, P (3, −2, −5) and Q (3, −2, 6), be PQ.
Since PQ passes through P (3, −2, −5), its position vector is given by,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7172/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m630936da.gif)
The direction ratios of PQ are given by,
(3 − 3) = 0, (−2 + 2) = 0, (6 + 5) = 11
The equation of the vector in the direction of PQ is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7172/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_36b135b3.gif)
The equation of PQ in vector form is given by, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7172/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_5277a5d3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7172/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_5277a5d3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7172/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m6fc8ce2d.gif)
The equation of PQ in Cartesian form is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7172/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_3f62e574.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7172/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_68de55ad.gif)
Question 10:
Find the angle between the following pairs of lines:
(i) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7173/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_7e018fd7.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7173/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_7e018fd7.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7173/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_2ea74592.gif)
(ii)
and
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7173/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_60ba656d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7173/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_4e01ef87.gif)
Answer:
(i) Let Q be the angle between the given lines.
The angle between the given pairs of lines is given by, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7173/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m10a04af.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7173/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m10a04af.gif)
The given lines are parallel to the vectors,
and
, respectively.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7173/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_90e6f6c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7173/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m57adec90.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7173/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_53ef8f73.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7173/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_61b3a329.gif)
(ii) The given lines are parallel to the vectors,
and
, respectively.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7173/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m6d1c8b55.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7173/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m869ef8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7173/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m181e6923.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7173/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_656e51bc.gif)
Question 11:
Find the angle between the following pairs of lines:
(i) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7175/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_1836c7f3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7175/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_1836c7f3.gif)
(ii) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7175/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m4919ebd4.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7175/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m4919ebd4.gif)
Answer:
- Let
and
be the vectors parallel to the pair of lines,
, respectively.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7175/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_18e6e30c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7175/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_6ecdbdd2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7175/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m18e8b3ce.gif)
The angle, Q, between the given pair of lines is given by the relation,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7175/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m50735be5.gif)
(ii) Let
be the vectors parallel to the given pair of lines,
and
, respectively.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7175/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_79b84703.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7175/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m64577a4a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7175/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m68f517d7.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7175/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_4e032e4c.gif)
If Q is the angle between the given pair of lines, then ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7175/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m10a04af.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7175/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m10a04af.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7175/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m76e6dd23.gif)
Question 12:
Find the values of p so the line
and
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7177/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m77f6d15.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7177/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m3d4dc83b.gif)
Answer:
The given equations can be written in the standard form as
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7177/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_11dee813.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7177/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_3614aa1f.gif)
The direction ratios of the lines are −3,
, 2 and
respectively.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7177/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_b6148f2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7177/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_c422fc8.gif)
Two lines with direction ratios, a1, b1, c1 and a2, b2, c2, are perpendicular to each other, if a1a2 + b1 b2 + c1c2 = 0
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7177/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_228d5f63.gif)
Thus, the value of p is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7177/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m33d87961.gif)
Question 13:
Show that the lines
and
are perpendicular to each other.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7178/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m1f31060b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7178/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_239c7a91.gif)
Answer:
The equations of the given lines are
and ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7178/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_239c7a91.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7178/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m1f31060b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7178/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_239c7a91.gif)
The direction ratios of the given lines are 7, −5, 1 and 1, 2, 3 respectively.
Two lines with direction ratios, a1, b1, c1 and a2, b2, c2, are perpendicular to each other, if a1a2 + b1 b2 + c1c2 = 0
∴ 7 × 1 + (−5) × 2 + 1 × 3
= 7 − 10 + 3
= 0
Therefore, the given lines are perpendicular to each other.
Question 14:
Find the shortest distance between the lines
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7179/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_mbb01f61.gif)
Answer:
The equations of the given lines are
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7179/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m42bac172.gif)
It is known that the shortest distance between the lines,
and
, is given by,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7179/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_288ace74.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7179/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m574ae0c2.gif)
d = b1→×b2→.a2→-a1→b1→×b2→
Comparing the given equations, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7179/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_4eb0d0c1.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7179/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_18a51ecd.gif)
Substituting all the values in equation (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7179/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_578cd9b9.gif)
Therefore, the shortest distance between the two lines is
units.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7179/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m67f80194.gif)
Question 15:
Find the shortest distance between the lines
and ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7181/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m708c15ab.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7181/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m575a126f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7181/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m708c15ab.gif)
Answer:
The given lines are
and ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7181/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m708c15ab.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7181/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m575a126f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7181/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m708c15ab.gif)
It is known that the shortest distance between the two lines,
, is given by,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7181/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m23bd385a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7181/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_7ed421d9.gif)
Comparing the given equations, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7181/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_5adaf041.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7181/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m4280200f.gif)
Substituting all the values in equation (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7181/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_1d11c41d.gif)
Since distance is always non-negative, the distance between the given lines is
units.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7181/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m3a7b7a85.gif)
Question 16:
Find the shortest distance between the lines whose vector equations are
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7182/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_446e29b3.gif)
Answer:
The given lines are
and ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7182/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m11f6b4a2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7182/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_2cc66c32.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7182/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m11f6b4a2.gif)
It is known that the shortest distance between the lines,
and
, is given by,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7182/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_288ace74.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7182/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m574ae0c2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7182/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_433f0024.gif)
Comparing the given equations with
and
, we obtain ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7182/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m66e7d813.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7182/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_288ace74.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7182/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m4e3b2763.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7182/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m66e7d813.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7182/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m60a3f97d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7182/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m1a2e4feb.gif)
Substituting all the values in equation (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7182/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m61a3dfde.gif)
Therefore, the shortest distance between the two given lines is
units.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7182/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m2751f62.gif)
Question 17:
Find the shortest distance between the lines whose vector equations are
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7185/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m7a0fcfbf.gif)
Answer:
The given lines are
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7185/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m4e1f1861.gif)
r→=(s+1)i^+(2s-1)j^-(2s+1)k^⇒r→=(i^-j^-k^)+s(i^+2j^-2k^) …(2)It is known that the shortest distance between the lines,
and
, is given by,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7185/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_288ace74.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7185/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m574ae0c2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7185/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m6e42d05c.gif)
For the given equations,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7185/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_2a0e89cf.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7185/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m470496ba.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7185/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_25f1e8a9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7185/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_1185bfb6.gif)
Substituting all the values in equation (3), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7185/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_5122a69d.gif)
Therefore, the shortest distance between the lines is
units.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/1/240/7185/NS_14-11-08_Gopal_12_Math_Exercise%2011.1_5_MNK_SS_html_m12abbd41.gif)