NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra Ex 10.4
Page No 454:
Question 1:
Find
, if
and
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7563/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m735c94a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7563/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_69f5f188.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7563/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m5173668d.gif)
Answer:
We have,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7563/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_69f5f188.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7563/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m5173668d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7563/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_bed85b6.gif)
Question 2:
Find a unit vector perpendicular to each of the vector
and
, where
and
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7565/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3c65b7a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7565/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_40c11246.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7565/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_14f2ee3a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7565/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_49059720.gif)
Answer:
We have,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7565/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_14f2ee3a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7565/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_49059720.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7565/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m5b71dd97.gif)
Hence, the unit vector perpendicular to each of the vectors
and
is given by the relation,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7565/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3c65b7a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7565/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_40c11246.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7565/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m751f7e4d.gif)
Question 3:
If a unit vector
makes an angles
with
with
and an acute angle θ with
, then find θ and hence, the compounds of
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_4f2bfef5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m4e8d241e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_3485de2a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_17396b03.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_4c61e793.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3ee0d69b.gif)
Answer:
Let unit vector
have (a1, a2, a3) components.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3ee0d69b.gif)
⇒ ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m2c8d461e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m2c8d461e.gif)
Since
is a unit vector,
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3ee0d69b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_46aae326.gif)
Also, it is given that
makes angles
with
with
, and an acute angle θ with![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_2131919c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3ee0d69b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m4e8d241e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_3485de2a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_17396b03.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_2131919c.gif)
Then, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m5c996f02.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3d7441d3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3ae32f08.gif)
Hence,
and the components of
are
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3984cf9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_4f2bfef5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7567/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_21ccf707.gif)
Question 4:
Show that
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7569/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_7d4c6cc4.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7569/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_6c5e506b.gif)
Question 5:
Find λ and μ if
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7571/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3f10f938.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7571/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m396f7587.gif)
On comparing the corresponding components, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7571/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_5908a1bb.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7571/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m66be8f0b.gif)
Hence, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7571/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m1349143c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7571/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m1349143c.gif)
Question 6:
Given that
and
. What can you conclude about the vectors
?
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7573/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_mba03bfc.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7573/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m4f84bf62.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7573/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_a1e644.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7573/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_mba03bfc.gif)
Then,
(i) Either
or
, or ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7573/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_mf69ec63.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7573/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_1150a7d5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7573/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_4fba4db2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7573/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_mf69ec63.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7573/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_3a511ee6.gif)
(ii) Either
or
, or ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7573/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_12be64de.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7573/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_1150a7d5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7573/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_4fba4db2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7573/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_12be64de.gif)
But,
and
cannot be perpendicular and parallel simultaneously.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7573/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_4f2bfef5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7573/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_6d43734a.gif)
Hence,
or
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7573/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_1150a7d5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7573/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_4fba4db2.gif)
Question 7:
Let the vectors
given as
. Then show that ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7575/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m2de91446.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7575/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m46f48cf5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7575/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_39b8445c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7575/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_7172503b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7575/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m2de91446.gif)
Answer:
We have,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7575/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_214916e5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7575/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m38f85818.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7575/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_2778d658.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7575/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_17898be0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7575/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m5a2e9ea0.gif)
On adding (2) and (3), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7575/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_35bc5314.gif)
Now, from (1) and (4), we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7575/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m1823f3df.gif)
Hence, the given result is proved.
Question 8:
If either
or
, then
. Is the converse true? Justify your answer with an example.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7578/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m4849a63d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7578/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m5f058d62.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7578/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_1c2e7367.gif)
Answer:
Take any parallel non-zero vectors so that
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7578/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_1c2e7367.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7578/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_409068f1.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7578/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_57bec8aa.gif)
It can now be observed that:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7578/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m9521a69.gif)
Hence, the converse of the given statement need not be true.
Question 9:
Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and
C (1, 5, 5).
Answer:
The vertices of triangle ABC are given as A (1, 1, 2), B (2, 3, 5), and
C (1, 5, 5).
The adjacent sides
and
of ΔABC are given as:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7580/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3aa9fe13.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7580/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_2fb90e8e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7580/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3aa9fe13.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7580/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3c4e982b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7580/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_2fb90e8e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7580/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m5a58bd22.gif)
Area of ΔABC ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7580/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m488c86f9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7580/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m488c86f9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7580/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_5f7aea0c.gif)
Hence, the area of ΔABC![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7580/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m17345ac1.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7580/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m17345ac1.gif)
Page No 455:
Question 10:
Find the area of the parallelogram whose adjacent sides are determined by the vector
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7582/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m64c6e9aa.gif)
Answer:
The area of the parallelogram whose adjacent sides are
is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7582/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_469c0fd8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7582/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_138c4299.gif)
Adjacent sides are given as:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7582/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m64c6e9aa.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7582/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m4285a934.gif)
Hence, the area of the given parallelogram is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7582/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_ff0cc59.gif)
Question 11:
Let the vectors
and
be such that
and
, then
is a unit vector, if the angle between
and
is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_4f2bfef5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m1a40228c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m1a446e7a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m746b3322.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_7aac94b3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m62d0504d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m1a40228c.gif)
(A)
(B)
(C)
(D) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m4c4df2e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m217ed0f1.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m6201c3cd.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m4e8d241e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m4c4df2e.gif)
Answer:
It is given that
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_223495ef.gif)
We know that
, where
is a unit vector perpendicular to both
and
and θ is the angle between
and
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m7b32c48.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_21845f56.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m62d0504d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m4670a45e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m62d0504d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m1a40228c.gif)
Now,
is a unit vector if
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_7aac94b3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3e2da210.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_7ee9f4d6.gif)
Hence,
is a unit vector if the angle between
and
is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_7aac94b3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m62d0504d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m1a40228c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7585/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m6201c3cd.gif)
The correct answer is B.
Question 12:
Area of a rectangle having vertices A, B, C, and D with position vectors
and
respectively is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7590/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m387b380d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7590/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_98c0fb9.gif)
(A)
(B) 1
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7590/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m5a4d85ce.gif)
(C) 2 (D) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7590/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m79b6bf4.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7590/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m79b6bf4.gif)
Answer:
The position vectors of vertices A, B, C, and D of rectangle ABCD are given as:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7590/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m2f3440f2.gif)
The adjacent sides
and
of the given rectangle are given as: ![](https://img-nm.mnimgs.com/img/study_content/content_ck_images/images/zafar1(1).png)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7590/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m27be3bd4.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7590/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_2fb90e8e.gif)
![](https://img-nm.mnimgs.com/img/study_content/content_ck_images/images/zafar1(1).png)
⇒AB→×BC→=2Now, it is known that the area of a parallelogram whose adjacent sides are
is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7590/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_469c0fd8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7590/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_138c4299.gif)
Hence, the area of the given rectangle is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7590/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m514783b1.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7590/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m514783b1.gif)
The correct answer is C.