The axes of coordinates are rotated about the z-axis through an angle of `pi//4` in the anticlockwise direction and the components of a vector are `2sqrt2, 3sqrt2, 4`. Prove that the components of the same vector in the original system are `-1, 5, 4`.

The axes are rotated through an angle pi//3 in the anticlockwise direction with respect to (0,0) . Find the coordinates of point (4,2) (w.r.t. old coordinate system) in the new coordinates system.

The point represented by the complex number (2-i) is rotated about origin through an angle of pi/2 in clockwise direction. The new position of the point will be-

If the lines represented by the equation 3y^2-x^2+2sqrt(3)x-3=0 are rotated about the point (sqrt(3),0) through an angle of 15^0 , one in clockwise direction and the other in anticlockwise direction, so that they become perpendicular, then the equation of the pair of lines in the new position is

A line segment has length 63 units and direction ratios are 3, -2, 6. The components of the line vector are-

If the pair of lines sqrt(3)x^2-4x y+sqrt(3)y^2=0 is rotated about the origin by pi/6 in the anticlockwise sense, then find the equation of the pair in the new position.

A particle P starts from the point z_0=1+2i , where i=sqrt(-1) . It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point z_1dot From z_1 the particle moves sqrt(2) units in the direction of the vector hat i+ hat j and then it moves through an angle pi/2 in anticlockwise direction on a circle with centre at origin, to reach a point z_2dot The point z_2 is given by (a) 6+7i (b) -7+6i (c) 7+6i (d) -6+7i

A unit vector of modulus 2 is equally inclined to x- and y-axes at an angle pi//3 . Find the length of projection of the vector on the z-axis.

If veca=2hati+5hatj+7hatk and vecb=hati-hatj+5hatk , then the vecor components of the vector 2veca-3vecb along the coordinate axes are -

Find the scalar and vector components of the vector with initial point (2,1) and terminal point (-5,7).

Find the position vector of the mid point of the vector joining the points P(2,3,4) and Q(4,1,-2).