`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(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_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.

If vec p is a unit vector and (vec x- vecp).(vec x+ vecp) = 8 then find |vec x|

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 , vec b are two non-collinear vectors, prove that the points with position vectors vec a+ vec b , vec a- vec b and vec a+lambda vec b are collinear for all real values of lambdadot

If vec(p) is a unit and (vec(x) - vec (p)) . (vec(x)+ vec (p)) = 80 then find |vec(x)|

Let vec(a),vec(b),vec (c ) be the positions vectors of the vertices of a triangle , prove that the area of the triangle is 1/2| vec(a) xx vec(b) + vec(b) xx vec(c)+ vec(c)xx vec(a)|

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 .

Let vec(a) = - hat(i) - hat(k) , vec(b) = - hat(i) +hat(j) and vec(c) = hat(i) + 2hat(j) + 3 hat(k) be three given vectors . If vec(r ) is a vector such that vec(r ) xx vec(b) = vec (c ) xx vec(b) and vec (r ) . vec (a ) = 0 then the value of vec(r ). vec(b) is -

Let vec a=- hat i- hat k , vec b=- hat i+ hat ja n d vec c= hat i+2 hat j+3 hat k be three given vectors. If vec r is a vector such that vec rxx vec b= vec cxx vec b and vec r dot vec a=0, then find the value of vec rdot vec bdot