The projection of `vec AB` on `vec CD` where `A(4,-3,2)`, `B(1,-1,1)`, `C(2,2,2)` and `D(3,3,3)` is

Find a unit vector in the direction of vec(AB) , where A(1, 2, 3) and B( 2, 3, 5) are the given points.

If vec a,vec b and vec c are the position vectors of the points A(2,3,-4),B(3,-4,-5) and C(3,2,-3) respectively,then |vec a+vec b+vec c| is equal to

If vec (a),vec (b) and vec (c) are the position vectors of the points A(2,3,-4), B(3,-4,-5) and C(3,2,-3) respectively ,then |vec(a)+vec(b)+vec(c)| is equal to (A) sqrt(113) (B) sqrt(185) (C) sqrt(203) (D) sqrt(209)

If A is (1,-2), B (3,k), C(-3,1) and D(k,4) where lines AB bot CD then :k=

Let vec a = a_1 hat i + a_2 hat j+ a_3 hat k;vec b = b_1 hat i+ b_2 hat j+ b_3 hat k ; vec c= c_1hat i + c_2 hat j+ c_3 hat k be three non-zero vectors such that vec c is a unit vector perpendicular to both vec a & vec b. If the angle between vec a and vec b is pi/6 , then |(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|^2=

i. Prove that the points vec a-2vec b+3vec c, 2vec a+3vec b-4vec c and -7vecb+10 vec c are are collinear, where vec a, vec b, vec c are non-coplanar. ii. Prove that the points A(1,2,3), B(3,4,7), and C(-3,-2,-5) are collinear. find the ratio in which point C divides AB.

A plane passes through the points A(1,2,3), B(2,3,1) and C(2,4,2) . If O is the origin and P is (2,-1,1) , then the projection of vec(OP) on the plane is of length :

Consider the following statements in respect of the vectors vec(u)_(1)= (1, 2, 3), vec(u)_(2)= (2, 3, 1) vec(u)_(3)= (1, 3,2) and vec(u)_(4)= (4,6,2) I vec(u)_(1) is parallel to vec(u)_(4) II. vec(u)_(2) is parallel to vec(u)_(4) III. vec(u)_(3) is parallel to vec(u)_(4) Which of the statements given above are correct?