Is a vector necessarily changed if it is rotated through anangle?

The vector hat i+x hat j+3 hat k is rotated through an angle theta and doubled in magnitude, then it becomes 4 hat i+(4x-2)dot hat j+2 hat k . Then value of x are -2/3 (b) 1/3 (c) 2/3 (d) 2

A vector is not changed if

Should the internal energy of a system necessarily increase if heat is added to it?

A uniform rod pivoted at its upper end hangs vertically. It is displaced through anangle of 60^@ and then released. Find the magnitude of the force acting on a particle of mass dm at the tip of the rod when the rod makes an angle of 37^@ with the vertical.

Should the internal energy of a system necessarily increase if its tempreture is increased?

Are f and g both necessarily onto, if gof is onto ?

A vector a has components a_1,a_2,a_3 in a right handed rectangular cartesian coordinate system OXYZ the coordinate axis is rotated about z axis through an angle pi/2 . The components of a in the new system

When a plane mirror is rotated through an angle theta , then the reflected ray turns through the angle 2theta , then the size of the image.

A vector has components p and 1 with respect to a rectangular Cartesian system. The axes are rotted through an angel alpha about the origin the anticlockwise sense. Statement 1: IF the vector has component p+2 and 1 with respect to the new system, then p=-1. Statement 2: Magnitude of the original vector and new vector remains the same.