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_3=p vec r_1+q vec r_2,` 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 Odot 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+dot+a_n=n b. a_1+a_2+dot+a_n=1 c. a_1+a_2+dot+a_n=0 d. a_1=a_2=a_3dot+a_n=0

If vec(PO) + vec(OQ) = vec(QO) + vec(OR) prove that the points P,Q,R are collinear .

Statement 1: Let vec a , vec b , vec ca n d vec d be the position vectors of four points A ,B ,Ca n dD and 3 vec a-2 vec b+5 vec c-6 vec d=0. Then points A ,B ,C ,a n dD are coplanar. Statement 2: Three non-zero, linearly dependent coinitial vector ( vec P Q , vec P Ra n d vec P S) are coplanar. Then vec P Q=lambda vec P R+mu vec P S ,w h e r elambdaa n dmu are scalars.

Statement 1: Let vec a , vec b , vec ca n d vec d be the position vectors of four points A ,B ,Ca n dD and 3 vec a-2 vec b+5 vec c-6 vec d=0. Then points A ,B ,C ,a n dD are coplanar. Statement 2: Three non-zero, linearly dependent coinitial vector ( vec P Q , vec P Ra n d vec P S) are coplanar. Then vec P Q=lambda vec P R+mu vec P S ,w h e r elambdaa n dmu are scalars.

If vec aa n d 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 bdot

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

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=ja n d 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.

If vec a , vec ba n d vec c are three non-zero vectors, no two of which ar collinear, vec a+2 vec b is collinear with vec c and vec b+3 vec c is collinear with vec a , then find the value of | vec a+2 vec b+6 vec c|dot

If vec(PO)+vec(OQ)=vec(QO)+vec(OR) , prove that the points P, Q,R, are collinear.