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Exercise 10.2 - Chapter 10 Vector Algebra class 12 ncert solutions Maths - SaraNextGen [2024]


Question 1:

Compute the magnitude of the following vectors:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7476/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_31caa58b.gif

Answer:

The given vectors are:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7476/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_59400371.gif

Question 2:

Write two different vectors having same magnitude.

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7479/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m77a7631e.gif

Hence, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7479/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_96f7d8b.gif are two different vectors having the same magnitude. The vectors are different because they have different directions.

Question 3:

Write two different vectors having same direction.

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7481/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m19981f5.gif

The direction cosines of https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7481/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_2b260d4b.gif are the same. Hence, the two vectors have the same direction.

Question 4:

Find the values of x and y so that the vectors https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7503/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_1e09f09f.gif are equal

Answer:

The two vectors https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7503/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_1e09f09f.gif will be equal if their corresponding components are equal.

Hence, the required values of x and y are 2 and 3 respectively.

Question 5:

Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (–5, 7).

Answer:

The vector with the initial point P (2, 1) and terminal point Q (–5, 7) can be given by,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7505/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3e8b1d95.gif

Hence, the required scalar components are –7 and 6 while the vector components are https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7505/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m7fe85b5f.gif

Question 6:

Find the sum of the vectorshttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7506/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m30282b9b.gif .

Answer:

The given vectors arehttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7506/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m30282b9b.gif .

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7506/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_1573fddd.gif

Question 7:

Find the unit vector in the direction of the vectorhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7508/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m2f1bc4fe.gif .

Answer:

The unit vector https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7508/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m4b61e9ff.gif in the direction of vector https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7508/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m2f1bc4fe.gif is given byhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7508/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m8220397.gif .

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7508/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_775fd640.gif

Question 8:

Find the unit vector in the direction of vectorhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7509/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_37573ad.gif , where P and Q are the points

(1, 2, 3) and (4, 5, 6), respectively.

Answer:

The given points are P (1, 2, 3) and Q (4, 5, 6).

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7509/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m240d73d1.gif

Hence, the unit vector in the direction of https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7509/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_37573ad.gif  is

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7509/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m66bb6ed6.gif .

Question 9:

For given vectors, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7511/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m79ed765d.gif and https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7511/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3fc656e2.gif , find the unit vector in the direction of the vector https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7511/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m7afd8463.gif

Answer:

The given vectors are https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7511/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m79ed765d.gif andhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7511/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3fc656e2.gif .

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7511/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m21d9071e.gif

Hence, the unit vector in the direction of https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7511/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_73889a02.gif is

a→+b→a→+b→=i^+k^2=12i⏜+12k⏜.

Question 10:

Find a vector in the direction of vector https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7513/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m72569e2.gif which has magnitude 8 units.

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7513/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_24386c6e.gif

Hence, the vector in the direction of vectorhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7513/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m72569e2.gif  which has magnitude 8 units is given by,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7513/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_b5d998e.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7513/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m649daf97.gif

Question 11:

Show that the vectorshttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7514/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m45d9d2dc.gif are collinear.

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7514/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m5bf8a3dd.gif .

Hence, the given vectors are collinear.

Question 12:

Find the direction cosines of the vector https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7516/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_1dda74be.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7516/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_mf23f50e.gif

Hence, the direction cosines of https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7516/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_3cb64f1d.gif

Question 13:

Find the direction cosines of the vector joining the points A (1, 2, –3) and

B (–1, –2, 1) directed from A to B.

Answer:

The given points are A (1, 2, –3) and B (–1, –2, 1).

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7518/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_3c8894e3.gif

Hence, the direction cosines of https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7518/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m27be3bd4.gif are https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7518/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m439a355c.gif

Question 14:

Show that the vector https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7519/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_26020955.gif is equally inclined to the axes OX, OY, and OZ.

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7519/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_2f6cde44.gif

Therefore, the direction cosines of https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7519/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_md243c2d.gif

Now, let αβ, and γbe the angles formed by https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7519/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_4f2bfef5.gif with the positive directions of xy, and z axes.

Then, we havehttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7519/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m2aa8c3f7.gif

Hence, the given vector is equally inclined to axes OX, OY, and OZ.

Question 15:

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7522/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_mc93ff0a.gif  respectively, in the ration 2:1

(i) internally

(ii) externally

Answer:

The position vector of point R dividing the line segment joining two points

P and Q in the ratio m: is given by:

1. Internally:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7522/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_b9eaa80.gif

2. Externally:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7522/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m30020ea7.gif

Position vectors of P and Q are given as:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7522/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_37b0a2c3.gif

(i) The position vector of point R which divides the line joining two points P and Q internally in the ratio 2:1 is given by,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7522/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m20bb3b45.gif

(ii) The position vector of point R which divides the line joining two points P and Q externally in the ratio 2:1 is given by,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7522/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_19a3f4d7.gif

Question 16:

Find the position vector of the mid point of the vector joining the points P (2, 3, 4) and Q (4, 1, – 2).

Answer:

The position vector of mid-point R of the vector joining points P (2, 3, 4) and Q (4, 1, – 2) is given by,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7524/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_7187c3d4.gif

Question 17:

Show that the points A, B and C with position vectors,https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7526/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_4abb5e40.gifhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7526/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_79bc6ffe.gif respectively form the vertices of a right angled triangle.

Answer:

Position vectors of points A, B, and C are respectively given as:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7526/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_mcc60c.gif  https://img-nm.mnimgs.com/img/study_content/content_ck_images/images/67368736482.png

AB→2+CA→2=35+6=41=BC→2Hence, ABC is a right-angled triangle.

Question 18:

In triangle ABC which of the following is not true:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7528/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_5e1b4db4.jpg

A. https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7528/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m2a9ed7f.gif

B. https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7528/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_2edf6a76.gif

C. https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7528/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_mb64d277.gif

D. https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7528/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_23234a70.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7528/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_5e1b4db4.jpg

On applying the triangle law of addition in the given triangle, we have:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7528/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m7b77e624.gif

From equations (1) and (3), we have:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7528/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m7a819ddd.gif

Hence, the equation given in alternative C is incorrect.

The correct Answer is C.

Question 19:

If https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_469c0fd8.gif are two collinear vectors, then which of the following are incorrect:

A. https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_354a54cd.gif , for some scalar λ

B. https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m626eecdc.gif

C. the respective components of https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_469c0fd8.gif are proportional

D. both the vectors https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_469c0fd8.gif have same direction, but different magnitudes

Answer:

If https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_469c0fd8.gif are two collinear vectors, then they are parallel.

Therefore, we have:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_354a54cd.gif  (For some scalar λ)

If λ = ±1, then https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m626eecdc.gif .

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_34d524ff.gif

Thus, the respective components of https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_469c0fd8.gif are proportional.

However, vectors https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_469c0fd8.gif can have different directions.

Hence, the statement given in D is incorrect.

The correct Answer is D.

Also Read : Exercise-10.3-Chapter-10-Vector-Algebra-class-12-ncert-solutions-Maths

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