If ` vec r_1, vec r_2, vec r_3` are the position vectors of the collinear points and scalar `p a n d q` exist such that ` vec r_1=p vec r_2+q vec r_3,` then show that `p+q=1.`

P(vec p) and Q(vec q) are the position vectors of two fixed points and R(vec r) is the position vectorvariable point. If R moves such that (vec r-vec p)xx(vec r -vec q)=0 then the locus of R is

Let vec r_1, vec r_2, vec r_3, .....vec r_n be the position vectors of points P_1,P_2, P_3 ,......P_n relative to the origin O . If the vector equation a_1 vec r_1+a_2 vec r_2+.....a_n vec r_n=0 hold, then a similar equation will also hold w.r.t. to any other origin provided a) a_1+a_2+......a_n=n b) a_1+a_2+....a_n=1 c) a_1+a_2+....a_n=0 d) a_1=a_2=a_3+a_n=0

If vec e_1, vec e_2, vec e_3a n d vec E_1, vec E_2, vec E_3 are two sets of vectors such that vec e_idot vec E_j=1,ifi=j and vec e_idot vec E_j=0a n difi!=j , then prove that [ (vec e_1, vec e_2 ,vec e_3)][ (vec E_1, vec E_2, vec E_3) ]=1.

Define the position vector of a point with respect to an origin. If vec(a) " and " vec(b) are the position vectors of the points P and Q respectively, find the vector vec(PQ) in terms of vec(a) " and " vec(b) .

If vec a and vec b are two given vectors and k is any scalar, then find the vector vec r satisfying vec rxx vec a+k vec r= vec b .

If vec ra n d vec s are non-zero constant vectors and the scalar b is chosen such that | vec r+b vec s| is minimum, then the value of |b vec s|^2+| vec r+b vec s|^2 is equal to a. 2| vec r|^2 b. | vec r|^2//2 c. 3| vec r""|^2 d. |r|^2

Let vec(a) " and " vec(b) be the position vectors of two given points P and Q respectively. To find the position vector of the point R which divides the line segment bar(PQ) internally in the ratio m:n.

If vec r. hat i= vec r.hat j= vec r.hat ka n d| vec r|=3, then find the vector vec rdot

If vec axx vec b= vec bxx vec c!=0,w h e r e vec a , vec b ,a n d vec c are coplanar vectors, then for some scalar k prove that vec a+ vec c=k vec bdot

Given three non-zero, non-coplanar vectors vec a , vec b ,and vecc .vec r_1=p vec a+q vec b+ vec ca n d vec r_2= vec a+p vec b+q vecc dot If the vectors vec r_1+2 vec r_2 and 2 vec r_1+ vec r_2 are collinear, then (p ,q) is