If a= i^+j^+k^,b=2i^−j^+3k^ and c=i^−2j^+k^find a unit vector parallel to the vector 2a−b+3c.
If a,b,andc be three non-coplanar vector and aprime,bprimeandc′ constitute the reciprocal system of vectors, then prove that r=(ra˙′)a+(rb˙′)b+(rc˙′)c r=(ra˙′)a′+(rb˙′)b′+(rc˙′)c′
Lett α,βandγ be distinct real numbers. The points whose position vector's are αi^+βj^+γk^;βi^+γj^+αk^andγi^+αj^+βk^
Let a,bandc be pairwise mutually perpendicular vectors, such that ∣a∣=1,∣∣b∣∣=2,∣c∣=2. Then find the length of a+b+c
Statement 1: ∣a∣=3,∣∣b∣∣=4and∣∣a+b∣∣=5,then∣∣a−b∣∣=5. Statement 2: The length of the diagonals of a rectangle is the same.
Points A(a),B(b),C(c)andD(d) are relates as xa+yb+zc+wd=0 and x+y+z+w=0,wherex,y,z,andw are scalars (sum of any two of x,y,znadw is not zero). Prove that if A,B,CandD are concylic, then ∣xy∣∣∣a−b∣∣2=∣wz∣∣∣c−d∣∣2˙
Statement 1: The direction cosines of one of the angular bisectors of two intersecting line having direction cosines as l1,m1,n1andl2,m2,n2 are proportional to l1+l2,m1+m2,n1+n2˙ Statement 2: The angle between the two intersection lines having direction cosines as l1,m1,n1andl2,m2,n2 is given by cosθ=l1l2+m1m2+n1n2˙
In triangle ABC,∠A=300,H
is the orthocenter and D
is the midpoint of BC.
is produced to T
such that HD=DT
The length AT
is equal to
(d). none of these