The vector `bar(i)+xbar(j)+3bar(k)` is rotated through an angle `theta` and doubled in magnitude,then it becomes `4bar(i)+(4x-2)bar(j)+2bar(k)` the value of x is

If the vector bar(i)-bar(j)+bar(k) bisects the angle between the vector bar(a) and the vector 4bar(i)+3bar(j) , then the unit vector in the direction of bar(a) is (xbar(i)+ybar(j)+zbar(k)) then x-15y+5z is equal to

If the vector -bar(i)+bar(j)-bar(k) bisects the angles between the vector bar(c) and the vector 3bar(i)+4bar(j) ,then the unit vector in the direction of bar(c) is

If lambda(2bar(i)-4bar(j)+4bar(k)) is a unit vector then lambda=

Let vec a be vector parallel to line of intersection of planes P_(1) and P_(2) through origin.If P_(1) is parallel to the vectors 2bar(j)+3bar(k) and 4bar(j)-3bar(k) and P_(2) is parallel to bar(j)-bar(k) and 3bar(I)+3bar(j), then the angle between vec a and 2bar(i)+bar(j)-2bar(k) is :

If the vectors bar(i)-2xbar(j)-3ybar(k) and bar(i)+3xbar(j)-2ybar(k) are orthogonal to each other,then the locus of the point (x,y) is

If the position vector of a point P is bar(r)=xbar(i)+ybar(j)+zbar(k), where x,y,z in N and bar(a) is a vector given by bar(a)=bar(i)+bar(j)+bar(k), then the totalnumber of possible positions of point P forwhich bar(r).bar(alpha)=10

If the vectors bar(a)=2bar(i)+3bar(j)+6bar(k) and bar(b) are collinear and |bar(b)|=21 , then bar(b) equals to

A unit vector perpendicular to the vector -bar(i)+2bar(j)+2bar(k) and making equal angles with x and y- axis can be

Let bar(OA)=bar(i)+2bar(j)+2bar(k) .In the plane bar(OA) and bar(i) rotate bar(OA) through 90^(@) about the origin O such that the new position of bar(OA) can be (A) (4bar(i)-bar(j)-bar(k))/(sqrt(2)) (B) (4bar(i)+2bar(j)+2bar(k))/(sqrt(2)) (C) 2bar(j)-2bar(k) (D) 6bar(i)-3bar(k)

The vector bar(b)=3bar(i)+4bar(k) is to be written as sum of a vector bar(b)_(1) parallel to bar(a)=bar(i)+bar(j) and a vector bar(b)_(2) perpendicular to bar(a) then bar(b)_(1) is equal to