Find the direction cosines of the vector joining the points A(1,2,3)andB(1,2,1), directed from A to B.
Find a vector magnitude 5 units, and parallel to the resultant of the vectors a=2i^+3j^−k^ and b=i^−2j^+k^˙
If AndB are two vectors and k any scalar quantity greater than zero, then prove that ∣∣A+B∣∣2≤(1+k)∣∣A∣∣2+(1+k1)∣∣B∣∣2˙
Let G1,G2andG3 be the centroids of the triangular faces OBC,OCAandOAB, respectively, of a tetrahedron OABC˙ If V1 denotes the volumes of the tetrahedron OABCandV2 that of the parallelepiped with OG1,OG2andOG3 as three concurrent edges, then prove that 4V1=9V1˙
If a,b,andc are mutually perpendicular vectors and a=α(a×b)+β(b×c)+γ(c×a)and[abc]=1, then find the value of α+β+γ˙
The number of distinct real values of λ , for which the vectors λ2i^+j^+k,i^−λ2j^+k^andi^+j^−λ2k^ are coplanar is a. zero b. one c. two d. three