When the axes are rotated through an angle $45^{\circ}$,the new coordinates of a point $P$ are $(1, -1)$. The coordinates of $P$ in the original system are

If the coordinates of a point $P$ change to $(2, -6)$ when the coordinate axes are rotated through an angle of $135^{\circ}$,then the coordinates of $P$ in the original system are

If the origin is shifted to $(1, -2)$ and the axes are rotated by an angle of $30^{\circ}$,what will be the new coordinates of $(1, 1)$?

The transformed equation of $3x^2 + 3y^2 + 2xy = 2$,when the coordinate axes are rotated through an angle of $45^{\circ}$,is

Suppose the axes $X$ and $Y$ are obtained by rotating the axes $x$ and $y$ by an angle $\theta$. If the equation $x^2+2 \sqrt{3} x y-y^2=4 a^2$ is transformed to $X^2-Y^2=2 a^2$ with respect to the $XY$-axes,then $\theta$ is equal to (in $^{\circ}$)

$A$ vector $\vec{a} = 2\hat{i} + 3\hat{j} + 7\hat{k}$ is given in a right-handed rectangular coordinate system. If the coordinate system is rotated about the $z-$axis from the positive $x-$axis to the positive $y-$axis through an angle of $\pi / 2$,then the new components of $\vec{a}$ will be: