# Rotate a vector with a quaternion @icon(box) @category(education) > Drag the angle and watch a vector turn in 3-D. A unit quaternion is the machine that does the turning — change the inputs and see it move. #vectors #quaternions #rotation /// **The one-line idea.** A quaternion is a compact way to store a rotation: *"spin by this angle about this axis."* Feed it a vector and it hands you back the rotated vector — no matrices, no gimbal lock. # Try it /// Pick how far to turn and which way the axis points. The vector below starts at $\hat x = \langle 1,0,0\rangle$ and the result updates live. angle = 90 // ° 0..360 axis_x = 0 // -1..1 axis_y = 0 // -1..1 axis_z = 1 // -1..1 @hero turned = rotate([1,0,0], [axis_x, axis_y, axis_z], angle) // @label(rotated x̂) /// The blue arrow (with arrowhead) is where $\hat x$ landed; the dashed grey arrow is where it started. Spin **angle** to 90° about the z-axis and $\hat x$ swings onto $\hat y = \langle 0,1,0\rangle$ — a clean quarter-turn. Now drag an **axis** slider and watch the whole rotation tilt out of the plane. @hero diagram = svg(``) # Why a quaternion can do this /// The trick lives in three imaginary units $i, j, k$ that multiply like the cross product: $\,i\cdot j = k$. Watch it directly: ij = quat(0,1,0,0) * quat(0,0,1,0) // @label(i · j) /// That $i\cdot j = k$ is the *same* right-hand rule as $\hat x \times \hat y = \hat z$. Because the algebra already encodes 3-D turning, one unit quaternion captures a whole rotation — and unlike Euler angles, it never locks up at the poles. # The rotation as a quaternion /// A turn by **angle** about a unit axis is stored as $q = \cos\tfrac\theta2 + \sin\tfrac\theta2\,(\text{axis})$. Here is the actual quaternion driving the arrow above: half = angle / 2 // @label(θ/2) q = quat(cos(half), sin(half)*axis_x, sin(half)*axis_y, sin(half)*axis_z) // @label(rotation quaternion) /// `rotate()` applies the sandwich product $q\,\vec v\,q^{-1}$ for you. Change the axis sliders to tilt the whole rotation in space — the quaternion, the arrow, and the result all follow.