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 b is not perpendicular to vec c , then find the vector vec r satisfying the equation vec rxx vec b= vec axx vec b and vec r. vec c=0.

If vec A and vec B are two vectors and k any scalar quantity greater than zero, then prove that | vec A+ vec B|^2lt=(1+k)| vec A|^2+(1+1/k)| vec B|^2dot

If c is a given non-zero scalar, and vec A and vec(B) are given non-zero vector such that vec A_|_ vecB , then find vector vec X which satisfies the equation vec A . vec X =c and vec Axx vec X= vec Bdot

If vec aa n d vec b are any two unit vectors, then find the greatest positive integer in the range of (3| vec a+ vec b|)/2+2| vec a- vec b| .

If vec a , vec b , and vec c are mutually perpendicular vectors of equal magnitudes, then find the angle between vectors vec a and vec a+ vec b+ vec c dot

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

Let vec a and vec b be unit vectors such that | vec a+ vec b|=sqrt(3) . Then find the value of (2 vec a+5 vec b).((3 vec a+ vec b+ vec axx vec b))dot

Let vec a , vec ba n d vec c be three non-coplanar vectors and vec p , vec qa n d vec r the vectors defined by the relation vec p=( vec bxx vec c)/([ vec a vec b vec c]), vec q=( vec cxx vec a)/([ vec a vec b vec c])a n d vec r=( vec axx vec b)/([ vec a vec b vec c])dot Then the value of the expression ( vec a+ vec b)dot vec p+( vec b+ vec c)dot vec q+( vec c+ vec a)dot vec r is a. 0 b. 1 c. 2 d. 3

Let vec A=2 vec i+ vec k , vec B= vec i+ vec j+ vec kdot vec C = 4hati-3hatj+7hatk Determine a vector vec R satisfying vec Rxx vec B= vec Cxx vec B and vec R. vec A=0.

Let vec a , vec ba n d vec c be three non-coplanar vecrors and vec r be any arbitrary vector. Then ( vec axx vec b)xx( vec rxx vec c)+( vec bxx vec c)xx( vec rxx vec a)+( vec cxx vec a)xx( vec rxx vec b) is always equal to [ vec a vec b vec c] vec r b. 2[ vec a vec b vec c] vec r c. 3[ vec a vec b vec c] vec r d. none of these