Let u^=u1i^+u2j^+u3k^ be a unit vector in be a unit vector in R3andw^=61(i^+j^+2k^).Given that there exists vector v^ in R3 such that ∣u^×v∣=1andw^.(u^×v)=1. Which of the following statement(s) is(are) correct?
ABCD is a tetrahedron and O is any point. If the lines joining O to the vrticfes meet the opposite faces at P,Q,RandS, prove that APOP+BQOQ+CROR+DSOS=1.
A pyramid with vertex at point P has a regular hexagonal base ABCDEF , Positive vector of points A and B are i^andi^+2j^ The centre of base has the position vector i^+j^+3k^˙ Altitude drawn from P on the base meets the diagonal AD at point G˙ find the all possible position vectors of G˙ It is given that the volume of the pyramid is 63 cubic units and AP is 5 units.
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.
Let D,EandF be the middle points of the sides BC,CAandAB, respectively of a triangle ABC˙ Then prove that AD+BE+CF=0 .
If a,b,candd are four vectors in three-dimensional space with the same initial point and such that 3a−2b+c−2d=0 , show that terminals A,B,CandD of these vectors are coplanar. Find the point at which ACandBD meet. Find the ratio in which P divides ACandBD˙
A man travelling towards east at 8km/h finds that the wind seems to blow directly from the north On doubling the speed, he finds that it appears to come from the north-east. Find the velocity of the wind.