NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra Miscellaneous Exercise
Page No 458:
Question 1:
Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.
Answer:
If
is a unit vector in the XY-plane, then ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7592/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m75c01749.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7592/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_41cbb03f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7592/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m75c01749.gif)
Here, θ is the angle made by the unit vector with the positive direction of the x-axis.
Therefore, for θ = 30°:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7592/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m4192e1af.gif)
Hence, the required unit vector is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7592/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m21af3b37.gif)
Question 2:
Find the scalar components and magnitude of the vector joining the points
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7594/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_131ca324.gif)
Answer:
The vector joining the points
can be obtained by,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7594/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_131ca324.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7594/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m111472e4.gif)
Hence, the scalar components and the magnitude of the vector joining the given points are respectively
and
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7594/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3e012ea6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7594/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_5e8c78e1.gif)
Question 3:
A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops. Determine the girl’s displacement from her initial point of departure.
Answer:
Let O and B be the initial and final positions of the girl respectively.
Then, the girl’s position can be shown as:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7597/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_32346bc6.jpg)
Now, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7597/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_7d133f98.gif)
By the triangle law of vector addition, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7597/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_32d7b395.gif)
Hence, the girl’s displacement from her initial point of departure is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7597/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m78dd8623.gif)
Question 4:
If
, then is it true that
? Justify your answer.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7600/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m5947c78d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7600/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_180ea095.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7600/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_mb09d08d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7600/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m1a58c7f9.jpg)
Now, by the triangle law of vector addition, we have
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7600/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m5947c78d.gif)
It is clearly known that
represent the sides of ΔABC.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7600/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3bda99f3.gif)
Also, it is known that the sum of the lengths of any two sides of a triangle is greater than the third side.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7600/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_5fba124.gif)
Hence, it is not true that
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7600/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_180ea095.gif)
Question 5:
Find the value of x for which
is a unit vector.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7602/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m8c6091.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7602/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m8c6091.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7602/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m516f96d4.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7602/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m63d9207d.gif)
Hence, the required value of x is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7602/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_ff0b8f0.gif)
Question 6:
Find a vector of magnitude 5 units, and parallel to the resultant of the vectors
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7604/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_4448472f.gif)
Answer:
We have,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7604/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_4448472f.gif)
Let
be the resultant of
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7604/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m66b53561.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7604/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/1/12/15/239/7604/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_298cc7e5.gif)
Hence, the vector of magnitude 5 units and parallel to the resultant of vectors
is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7604/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/1/12/15/239/7604/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m37556287.gif)
Question 7:
If
, find a unit vector parallel to the vector
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7606/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m6f16a1b6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7606/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_1d46e5e5.gif)
Answer:
We have,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7606/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m6f16a1b6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7606/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m58dcc88c.gif)
Hence, the unit vector along
is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7606/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_1d46e5e5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7606/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_78a10c20.gif)
Question 8:
Show that the points A (1, –2, –8), B (5, 0, –2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.
Answer:
The given points are A (1, –2, –8), B (5, 0, –2), and C (11, 3, 7).
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7609/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_4c4143d6.gif)
Thus, the given points A, B, and C are collinear.
Now, let point B divide AC in the ratio
. Then, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7609/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m78955cf8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7609/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_73d98b55.gif)
On equating the corresponding components, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7609/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_7530303e.gif)
Hence, point B divides AC in the ratio![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7609/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_40c1d64c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7609/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_40c1d64c.gif)
Question 9:
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are
externally in the ratio 1: 2. Also, show that P is the mid point of the line segment RQ.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7611/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3671c5d7.gif)
Answer:
It is given that
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7611/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_4da6b64.gif)
It is given that point R divides a line segment joining two points P and Q externally in the ratio 1: 2. Then, on using the section formula, we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7611/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_56865306.gif)
Therefore, the position vector of point R is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7611/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_1c729e19.gif)
Position vector of the mid-point of RQ =![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7611/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m67bf753c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7611/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m67bf753c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7611/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_4df158a.gif)
Hence, P is the mid-point of the line segment RQ.
Question 10:
The two adjacent sides of a parallelogram are
and
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7613/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m63343167.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7613/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_23061c60.gif)
Find the unit vector parallel to its diagonal. Also, find its area.
Answer:
Adjacent sides of a parallelogram are given as:
and![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7613/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_462bc0ac.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7613/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_1049dce5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7613/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_462bc0ac.gif)
Then, the diagonal of a parallelogram is given by
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7613/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m7afd8463.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7613/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_4b2ae18.gif)
Thus, the unit vector parallel to the diagonal is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7613/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m22174ddf.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7613/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m235be74d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7613/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_4dd19828.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7613/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m116f4421.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7613/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3b11431d.gif)
Hence, the area of the parallelogram is
square units.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7613/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_34ef70c0.gif)
Question 11:
Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7615/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m1a80faef.gif)
Answer:
Let a vector be equally inclined to axes OX, OY, and OZ at angle α.
Then, the direction cosines of the vector are cos α, cos α, and cos α.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7615/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m561a3303.gif)
Hence, the direction cosines of the vector which are equally inclined to the axes are
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7615/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m1a80faef.gif)
Question 12:
Let
and
. Find a vector
which is perpendicular to both
and
, and
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7620/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m7c7e0571.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7620/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_mece6e8d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7620/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m765a894b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7620/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/1/12/15/239/7620/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/1/12/15/239/7620/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m63077624.gif)
Answer:
Let
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7620/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m75613acd.gif)
Since
is perpendicular to both
and
, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7620/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m42cf08f4.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7620/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/1/12/15/239/7620/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_6d43734a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7620/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_52f0ac0b.gif)
Also, it is given that:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7620/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m5fa5d036.gif)
On solving (i), (ii), and (iii), we get:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7620/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_f46c9fa.gif)
Hence, the required vector is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7620/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_1d51b915.gif)
Question 13:
The scalar product of the vector
with a unit vector along the sum of vectors
and
is equal to one. Find the value of
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7623/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_26020955.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7623/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_6dc5faed.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7623/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_2d8aac5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7623/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m11cc021f.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7623/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_6bebeef1.gif)
Therefore, unit vector along
is given as:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7623/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m60d538d6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7623/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3d85b034.gif)
Scalar product of
with this unit vector is 1.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7623/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_4e452768.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7623/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m7436605.gif)
Hence, the value of λ is 1.
Question 14:
If
are mutually perpendicular vectors of equal magnitudes, show that the vector
is equally inclined to
and
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7627/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_6c7fdab8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7627/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_78a9edac.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7627/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m4100a78d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7627/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_73cf45c2.gif)
Answer:
Since
are mutually perpendicular vectors, we have
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7627/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_13196ed7.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7627/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_70cb4005.gif)
It is given that:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7627/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m6779e2dd.gif)
Let vector
be inclined to
at angles
respectively.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7627/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_27b1e586.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7627/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_13196ed7.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7627/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m4c746ce8.gif)
Then, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7627/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m1abd2fac.gif)
Now, as
,
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7627/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m6779e2dd.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7627/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m6b12bc3d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7627/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m1f37cf43.gif)
Hence, the vector
is equally inclined to
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7627/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m7a318cb2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7627/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_13196ed7.gif)
Page No 459:
Question 15:
Prove that
, if and only if
are perpendicular, given
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7628/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m79735266.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7628/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m290fce70.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7628/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m14f2eb22.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7628/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_3311e874.gif)
Question 16:
If θ is the angle between two vectors
and
, then
only when
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7630/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/1/12/15/239/7630/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/1/12/15/239/7630/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m9cb2a72.gif)
(A)
(B) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7630/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m7767fc1b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7630/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_78a274fd.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7630/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m7767fc1b.gif)
(C)
(D) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7630/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_374b5b60.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7630/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_6cccca2c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7630/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_374b5b60.gif)
Answer:
Let θ be the angle between two vectors
and
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7630/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/1/12/15/239/7630/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m4670a45e.gif)
Then, without loss of generality,
and
are non-zero vectors so that
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7630/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/1/12/15/239/7630/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/1/12/15/239/7630/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_64088af0.gif)
It is known that
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7630/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_mf1b0571.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7630/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_31c2b1c1.gif)
Hence,
when
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7630/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m9cb2a72.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7630/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m1b29b425.gif)
The correct answer is B.
Question 17:
Let
and
be two unit vectors andθ is the angle between them. Then
is a unit vector if
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7632/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/1/12/15/239/7632/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/1/12/15/239/7632/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3c65b7a.gif)
(A)
(B)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7632/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_2956c37.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7632/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_4c7073d2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7632/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m20e33c8d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7632/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_666fdc32.gif)
Answer:
Let
and
be two unit vectors andθ be the angle between them.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7632/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/1/12/15/239/7632/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m4670a45e.gif)
Then,
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7632/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m160e97b3.gif)
Now,
is a unit vector if
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7632/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/1/12/15/239/7632/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_67f54729.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7632/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_795d5423.gif)
Hence,
is a unit vector if
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7632/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/1/12/15/239/7632/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_9d761b9.gif)
The correct answer is D.
Question 18:
The value of
is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7633/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3420ad89.gif)
(A) 0 (B) –1 (C) 1 (D) 3
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7633/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m57d565ee.gif)
The correct answer is C.
Question 19:
If θ is the angle between any two vectors
and
, then
when θisequal to
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7635/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/1/12/15/239/7635/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/1/12/15/239/7635/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_29f5d65c.gif)
(A) 0 (B)
(C)
(D) π
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7635/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_3008de06.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7635/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m1b6d011a.gif)
Answer:
Let θ be the angle between two vectors
and
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7635/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/1/12/15/239/7635/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m4670a45e.gif)
Then, without loss of generality,
and
are non-zero vectors, so that
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7635/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/1/12/15/239/7635/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/1/12/15/239/7635/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_64088af0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7635/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m12be0e6f.gif)
Hence,
when θisequal to
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7635/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_29f5d65c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7635/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m6201c3cd.gif)
The correct answer is B.