Let a,b,c be distinct non-negative numbers and the vectors ai^+aj^+ck^,i^+k^,ci^+cj^+bk^ lie in a plane, and then prove that the quadratic equation ax2+2cx+b=0 has equal roots.
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˙
Prove that the four points 6i^−7j^,16i^−19j^−4k^,3j^−6k^and2i^+5j^+105^ form a tetrahedron in space.
Statement 1: If cosα,cosβ,andcosγ are the direction cosines of any line segment, then cos2α+cos2β+cos2γ=1. Statement 2: If cosα,cosβ,andcosγ are the direction cosines of any line segment, then cos2α+cos2β+cos2γ=1.
If a,b are two non-collinear vectors, prove that the points with position vectors a+b,a−b and a+λb are collinear for all real values of λ˙
Find the values of λ such that x,y,z=(0,0,0)and(i^+j^+3k^)x+(3i^−3j^+k^)y+(−4i^+5j^)z=λ(xi^+yj^+zk^), where i^,j^,k^ are unit vector along coordinate axes.