Class name Subject Book title Chapter number Content Created at
Class XII Mathematics Mathematics Part-II 4 1 2 2 3 3 (iii) the vectors are equal if and only if a = b , a = b and a = b 1 1 2 2 3 3 (iv) the multiplication of vector by any scalar λ is given...
Class XII Mathematics Mathematics Part-II 4 are equal. Thus, the given vectors will be equal if and only if x = 2, y = 2, z = 1 Reprint 2026-27 350 MATHEMATICS Example 5 Let and . Is ? Are th...
Class XII Mathematics Mathematics Part-II 4 and n are the direction cosines of the given vector, then Thus, the direction cosines are  1 , 1 ,– 2  .  6 6 6  10.5.2 Vector joining two poin...
Class XII Mathematics Mathematics Part-II 4 n are positive scalars, we say that the point R divides internally in the ratio of m : n. Now from triangles ORQ and OPR, we have = and = , Therefo...
Class XII Mathematics Mathematics Part-II 4 Computethemagnitudeofthefollowingvectors: = ˆ ˆ = ˆ ˆ ˆ = 1 i + 1 j − 1 k i + j + k; 2i −7 j −3k; 3 3 3 2. Write two different vectors having same ...
Class XII Mathematics Mathematics Part-II 4 ALGEBRA 355 16. Findthepositionvectorofthemidpointofthevectorjoiningthepoints P(2,3,4) and Q(4, 1, –2). 17. Show that the points A, B and C with po...
Class XII Mathematics Mathematics Part-II 4 ⋅ = −| || | In particular, , as θ in this case is π. 5. In view of the Observations 2 and 3, for mutually perpendicular unit vectors i, j and k, we...
Class XII Mathematics Mathematics Part-II 4 positive x-axistowardthepositive y-axis(Fig10.22(ii)). Fig 10.22 (i), (ii) Definition 3 The vector product of two nonzero vectors , is denoted by a...
Class XII Mathematics Mathematics Part-II 4 l (say), in the anticlockwise direction (Fig 10.20). Then the projection of on l is a vector (say)withmagnitude ,andthedirectionof beingthesame(oro...
Class XII Mathematics Mathematics Part-II 4 when =1. Solution Given . We have Reprint 2026-27 VECTOR ALGEBRA 359 Example 14 Find angle ‘θ’ between the vectors . Solution The angle θ between t...
Class XII Mathematics Mathematics Part-II 4 have . So, let us assume that . Then, we have = |cosθ|≤1 Therefore C Example 20 For any two vectors , we always a + b have (triangleinequality). b ...
Class XII Mathematics Mathematics Part-II 4 7 7 7 Also, show that they are mutually perpendicular to each other. Reprint 2026-27 362 MATHEMATICS 6. Find , if . 7. Evaluate the product . 8. Fi...
Class XII Mathematics Mathematics Part-II 4 j. ˆ 8. If represent the adjacent sides of a triangle then its area is given as . Bydefinitionoftheareaofatriangle,wehavefrom Fig10.26, Area of tri...
Class XII Mathematics Mathematics Part-II 4 k × j = − j×k) ˆ ˆ ˆ ˆ = a b k −a b j −a bk +a b i + a b j −a b i ˆ ˆ 1 2 1 3 2 1 2 3 3 1 3 2 (as i × j = k, j×k =i and k×i = j) ˆ ˆ ˆ = (a 2 3a b ...
Class XII Mathematics Mathematics Part-II 4 is 21 Reprint 2026-27 368 MATHEMATICS Example 25 Find the area of a parallelogram whose adjacent sides are given by the vectors Solution The area o...
Class XII Mathematics Mathematics Part-II 4 (D) 4 Miscellaneous Examples Example 26 Write all the unit vectors in XY-plane. Solution Let be a unit vector in XY-plane (Fig 10.28). Then, from t...
Class XII Mathematics Mathematics Part-II 4 29 Three vectors satisfy the condition . Evaluate the quantity . Solution Since , we have = 0 or = 0 Therefore ... (1) Again, = 0 or ... (2) Simila...
Class XII Mathematics Mathematics Part-II 4 position vectors are externally in the ratio 1 : 2.Also, show that P is the mid point of the line segment RQ. 10. The two adjacent sides of a paral...
Class XII Mathematics Mathematics Part-II 4 z)isgivenas ,andits ® magnitudeby x + y + z2 . ® The scalar components of a vector are its direction ratios, and represent its projections along th...
Class XII Mathematics Mathematics Part-II 4 )3 ˆ; = a1 1+ a 2 2 a b3 3; ˆ ˆ ˆ i j k and = a b c . 1 1 1 a2 b 2 c2 Historical Note The word vector has been derived from a Latin word vectus, wh...
Class XII Mathematics Mathematics Part-II 4 to the D. Heaviside and P.G. Tait (1831-1901) whocontributedsignificantlytothissubject. — ❖ — Reprint 2026-27
Class XII Mathematics Mathematics Part-II 5 THREE DIMENSIONAL GEOMETRY 377 Chapter 11 THREE DIMENSIONAL GEOMETRY ❖ The moving power of mathematical invention is not reasoning but imagination....
Class XII Mathematics Mathematics Part-II 5 cosines are denoted by l, m and n. RemarkIfthegivenlineinspacedoesnotpassthroughtheorigin,then,inordertofind its direction cosines, we draw a line ...
Class XII Mathematics Mathematics Part-II 5 Since one and only one line passes through two given points, we can determine the directioncosinesofalinepassingthroughthegivenpointsP(x ,y , z )an...
Class XII Mathematics Mathematics Part-II 5 + (−2) 2 2 + (−1) + (−2) 2 2 + −1 + (−2) 2 2 ,1 −2, or 3 3 3 Example 3 Find the direction cosines of the line passing through the two points (– 2, ...