Find a unit vector `vec c` if `vec -i+vec j-vec k` bisects the angle between `vec c and 3 vec i+4vec j`.

If the vector vec a= x vec i + y vec j + 2 vec k is perpendicular to the vector vec b = vec i - vec j + vec k and the scalar product of vec a and vec c= vec i + 2 vec j is equal to 4 then x= ________ and y=__________.

Find the unit vector in the direction of the vector vec a + vec b if vec a =vec i +2 vec j + 3 vec k and vec b = 2 vec i + 3 vec j + 5 vec k .

The vector vec x directed along the bisector of the angle between the vector vec a=7vec i-4vec j-4vec k and vec b=-2vec i-vec j+2vec k if |x|=45 is

Find the value of: (vec j xxvec k) * (3vec i + 2vec j-vec k)

If vec a = vec i + vec j + vec k, vec b = vec i - vec j + vec k and vec c = vec i + 2vec j - vec k, then the value of | [vec a.vec a, vec a.vec b, vec a.vec c], [vec b.vec a, vec b.vec b, vec b.vec c], [vec c.vec a, vec c.vec b, vec c.vec c ] | is equal to: (1) 2 (2) 4 (3) 16 (4) 64

The unit vector along vec(i)+vec(j) is :-

If a vector vec r of magnitude 3sqrt(6) is directed along the bisector of the angle between the vectors vec a=7vec i-4vec j-4vec k and vec b=-2vec i-vec j+2vec k, then vec r is equal to

Vector of length of 3 unit which is perpendicular to vec i + vec j + vec k and lies in the plane of vec i + vec j + vec k and 2vec i-3vec j, is (1) (3) / (sqrt ( 6)) (vec i-2vec j + vec k) (2) (3) / (sqrt (6)) (2vec i-vec j-vec k) (3) (3) / (sqrt (114)) ( 8vec i-7vec j-vec k) (4) (3) / (sqrt (114)) (- 7vec i + 8vec j-vec k)

The position vectors of the points A,B,C , D are respectively 2 vec i + vec j - vec k, vec i + vec j + vec k, vec i- 2 vec j+ 3 vec k and 3 vec i- vec j + 2 vec k . Evaluate [vec (AB), vec (AC), vec (AD)] .

Let ABCD be the parallelogram whose sides AB and AD are represented by the vectors 2vec i+4vec j-5vec k and vec i+2vec j+3vec k respectively. Then if vec a is a unit vector parallel to vec AC ,then vec a is equal to