Why Do Rotation Matrices About the y-Axis Have Different Signs?

The signs in a rotation matrix depend on what is being rotated and how vectors are represented. With the usual convention—a right-handed coordinate system, column vectors, and an active rotation of the vector—the rotation by angle $\theta$ about the positive $y$-axis is

$$R_y(\theta)= \begin{pmatrix} \cos\theta & 0 & \sin\theta\\ 0 & 1 & 0\\ -\sin\theta & 0 & \cos\theta \end{pmatrix}.$$

The most useful way to check the signs is to apply the matrix to a coordinate-axis vector and use the right-hand rule. If you see the opposite signs in a book or software library, check whether it is describing a passive coordinate transformation or using row vectors.

What the standard matrix does

An active rotation moves a vector while keeping the coordinate axes fixed. For a column vector

$$\mathbf v=\begin{pmatrix}x\\y\\z\end{pmatrix},$$

multiplication gives

$$R_y(\theta)\mathbf v= \begin{pmatrix} x\cos\theta+z\sin\theta\\ y\\ -x\sin\theta+z\cos\theta \end{pmatrix}.$$

The $y$-coordinate stays the same because the vector rotates around the $y$-axis. The $x$ and $z$ coordinates mix, just as the $x$ and $y$ coordinates mix in a rotation in the plane.

Why the signs have this order

In a right-handed coordinate system, positive rotation about an axis follows the right-hand rule: point your right thumb along the positive axis, and your curled fingers show the direction of positive rotation. For the positive $y$-axis, a vector initially pointing along positive $x$ turns toward negative $z$.

That gives a quick sign check. Set $\theta=90^\circ$ and apply the matrix to the positive $x$-direction:

$$R_y(90^\circ) \begin{pmatrix}1\\0\\0\end{pmatrix} = \begin{pmatrix}0\\0\\-1\end{pmatrix}.$$

So positive $x$ has moved to negative $z$, as the right-hand rule requires. Similarly, the positive $z$-direction moves to positive $x$:

$$R_y(90^\circ) \begin{pmatrix}0\\0\\1\end{pmatrix} = \begin{pmatrix}1\\0\\0\end{pmatrix}.$$

These checks are often clearer than trying to remember which off-diagonal entry gets a plus sign.

Why another source may show the opposite signs

You may also encounter

$$\begin{pmatrix} \cos\theta & 0 & -\sin\theta\\ 0 & 1 & 0\\ \sin\theta & 0 & \cos\theta \end{pmatrix}.$$

This is $R_y(-\theta)$, the inverse (or transpose) of the standard active rotation matrix. It commonly appears when the axes or coordinate frame are rotated while the geometric vector is held fixed. That is a passive transformation: the vector’s coordinates change in the opposite direction to the active rotation.

The same transpose appears if a convention represents vectors as rows rather than columns. With row vectors, you multiply on the right, so the matrix used for the same geometric transformation is transposed. Some applications also choose a left-handed coordinate system or define positive angles differently. Always identify the convention before comparing signs.

Common mistakes

• Using the right-hand rule from the wrong end of the axis. Imagine looking along the positive axis toward the origin, or use the thumb-and-fingers rule consistently; reversing the viewing direction can make the apparent turn look opposite.
• Mixing active and passive rotations. Rotating the object by $\theta$ and rotating the coordinate frame by $\theta$ produce inverse coordinate operations.
• Forgetting the vector convention. A matrix acting on column vectors from the left and a matrix acting on row vectors from the right are transposes for the same transformation.
• Checking only the cosine entries. The diagonal terms are the same for $\theta$ and $-\theta$; the sine terms reveal the sign convention.

If you are implementing a rotation, test it on a basis vector at $90^\circ$. That catches a sign or convention mismatch immediately.