E.g. Question 7 of Tutorial 1 Show that the vectors, and represents an isosceles triangle. Solution Let, and They represents a triangle. Hence, an isosceles triangle.

E.g. Question 8 of Tutorial 1 Using the vectors, show that the diagonals of the quadrilateral OABC bisect each other if and only if OABC is a parallelogram. Solution Suppose that the diagonals of the quadrilateral ABCD bisect each other. O A B C D Since diagonals bisect OD = DB AD = DC Hence, OABC is a parallelogram. Similarly show that other way also true. Please, work out all the questions in Tute One and Two in holidays.

Dot product. For two vectors and Dot product is, where is the angle between the vectors. ;s;a.=Ks;h.=Ks;h iuSnkaO m%;sM, / Properties hehs.ksuq

ys idOkh 1 fuúg iEu m%ldYkhlau Y+kH jk fyhska m%;sM,h i;H fjS" 2 fuúg iy yd osYd iudk iy fuS ksid m%;sM,h i;H fjS" m%:ufhka nj fmkajuq"

3 fuúg iy ys osYd m%;súreoO fjS" fuS ksid m%;sM,h i;H fjS" tkuS

E.g. Find the angle between the following vectors. Solution hehs.ksuq

l;sr.=Ks;h iuSnkaO m%;sM, You have access to these slides at the computer centre, Dept. of Math. Please, refer them. E.g. Find the cross product of the following vectors.

E.g. Find the cross product of the following vectors.

Definition of the dot and cross products. For two vectors and Dot product is, where is the angle between the vectors. Cross product is, where is the angle between the vectors. is u unit vector such that is a right handed system. Now, we want to connect three vectors using only dot product and / or cross product. How we can do it ?

;%s;aj.=Ks;h ' / Triple Product. ffoYsl ;=kla w¾:j;a f,i ;s;a.=Ks;h iy fyda l;sr.=Ks;h iu. fhoSfuka ;%s;aj.=Ks; w¾: olajkq,nhs. The triple products are defined using dot product and cross product meaningfully with three vectors. ;%s;aj.=Ks; / Triple Product ;%s;aj wosY.=Ks;h / Triple scalar product. ;%s;aj ffoYsl.=Ks;h / Triple vector product.

;%s;aj wosY.=Ks;h / Triple scalar product. ffoYsl ;=ku tlu,laIHfhka wdruSN jk f,i w|suq' OA C D fujeks ;%sudk rEmhlg iudka;rKslrKh ( parallelopipe) hkak Ndú; fjS" ABCD

OA C D iudka;rKslrKfha mrsudj = ABCD. c ;jo tkuS ABCD = iudka;rKslrKfha mrsudj fus ksid ;jo

;%s;aj ffoYsl.=Ks;h / Triple vector product. X Y Z

Eq 1 Eq 2 So

E.g. Evaluate Let

m%fuÞhh AB f¾Ld LKavh C,laIHfhka wkqmd;hg wNHka;rj fnfoa. O uQ,hla wkqnoaOfhka fjÞ. fuys A B C O Theorem The point C divides the line segment internally to the ratio. With respect to an origin O,. Here.

m%fuÞhh A,B iy C,laIH talf¾Çh kuS yd kuSu muKla jk f,i tl úg Y+kH fkdjQ iy wosY f,i mj;S. fuys Theorem The points A, B and C are collinear iff there exist scalars λ, μ and γ, non-zero at the same time such that and. Proof/ idOkh A,B iy C,laIH talf¾Çh kuS A B C O Choose Then we have the given conditions.

iy úg A,laIHfhka BC f¾Ldj wkqmd;hg fnfoa" A,B iy C,laIH talf¾Çh fjS. The point A divides the line BC to the ratio γ : μ. So, A, B and C are collinear. E.g. Use vectors, to show that the diagonals of a parallelogram bisect. ffoYsl Ndú;fhka iudka;rdi%hl úl¾K iupSfPAokh jk nj fmkajkak. E.g. Find the ratio of intersection of the medians of a triangle. ffoYsl Ndú;fhka ;%sflaKhl uOHia: fPaokh jk wkqmd;h fidhkak.

E.g. Find the ratio of intersection of the medians of a triangle. ksoiqk ffoYsl Ndú;fhka ;%sflaKhl uOHia: fPaokh jk wkqmd;h fidhkak. A B C s 1 t 1 D E ABC ;s%fldaKhg m%fuÞhh fhoSfuka F ABE ;%sfldaKhg m%fuÞhh fhoSfuka fuS ksid wkqrEm ffoYslhkays ix.=Kl ie,lSfuka 1 2

1 2 1 uOHia: 1:2 wkqmd;hg fPaokh fjS" A B C s 1 D E F kuS