P is the point `vec(i)+x vec(j)+3vec(k)`. The vector `bar(OP)` ('O' is the origin) is rotated about the point 'O' through an angle `theta`. Q is point `4vec(i)+(4x-2)vec(j)+2vec(k)` on the new support of `bar(OP)` such that `OQ=2OP`. Then x value is

If vec a xx i+2vec a-4vec i-2vec j+vec k=0 then vec a=

If the projection of vec(i)+3vec(j)+7vec(k) on 2vec(i)-3vec(j)+6vec(k) is ___________.

If the vectors 2vec(i)-qvec(j)+3vec(k) and 4vec(i)-5vec(j)+6vec(k) are collinear, the value of q is

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

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

The perpendicular distance from origin tot plane passing through the points 2bar(i)-2bar(j)+bar(k),3vec i+2bar(j)-bar(k),3vec i-bar(j)-2bar(k) is

Statement 1: let A( vec i+ vec j+ vec k)a n dB( vec i- vec j+ vec k) be two points. Then point P(2 vec i+3 vec j+ vec k) lies exterior to the sphere with A B as its diameter. Statement 2: If Aa n dB are any two points and P is a point in space such that vec P Adot vec P B >0 , then point P lies exterior to the sphere with A B as its diameter.

The position vector of a midpoint lying on the line joining the points whose position vectors are vec i+vec j-vec k and vec i-vec j+vec k is

The sine of the angle between the vectors vec i+3vec j+2vec k and 2vec i-4vec j+vec k is

A point on the line r=2vec i+3vec j+4vec k+t(vec i+vec j+vec k) is