Unit Tangent Vector Calculator — T(t) with Steps

这个单位切向量计算器对分量为 t 的多项式的曲线 r(t) = ⟨x(t), y(t), z(t)⟩ 求 T(t) = r′(t)/|r′(t)|。它对每个分量求导，求出 r′(t) 的长度，并在你选择的 t 值处精确给出 T——例如 ⟨1/3, 2/3, 2/3⟩——出现根式时同时给出小数。

公式

\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{\lVert \mathbf{r}'(t) \rVert}

Divide the velocity vector by its length so the result has length 1.

\lVert \mathbf{r}'(t) \rVert = \sqrt{x'(t)^2 + y'(t)^2 + z'(t)^2}

The length (speed) of r′(t).

Choose a 2D or 3D curve.
Type each component as a polynomial in t, such as t, t^2 and 2t^3/3.
Enter the value of t and press Calculate: you get T at that point, the speed |r′(t)|, and the general formula for T(t).

r′(t) = ⟨3, 4 − 2t⟩, so r′(0) = ⟨3, 4⟩.

|r′(0)| = √(9 + 16) = 5.

T(0) = ⟨3, 4⟩/5 = ⟨3/5, 4/5⟩.

r′(t) = ⟨1, 2t, 2t²⟩, so r′(1) = ⟨1, 2, 2⟩.

|r′(1)| = √(1 + 4 + 4) = 3.

T(1) = ⟨1/3, 2/3, 2/3⟩.

r′(t) = ⟨2t, 3t²⟩, so r′(1) = ⟨2, 3⟩.

|r′(1)| = √(4 + 9) = √13.

T(1) = ⟨2/√13, 3/√13⟩ = ⟨(2√13)/13, (3√13)/13⟩ ≈ ⟨0.5547, 0.8321⟩.

Dividing r(t) by its length instead of r′(t): the tangent comes from the derivative.
Forgetting to square each component before adding under the square root.
Trying to find T where r′(t) = 0: the curve has no defined direction there.
Leaving out the z-component of a 3D curve when finding the length.

T(t) is the direction of motion along a curve r(t), scaled to length 1: T(t) = r′(t)/|r′(t)|. It points the way the curve is heading at each t.

Differentiate each component to get r′(t), substitute the value of t, find the length of that vector, and divide each component by the length.

Not yet — each component must be a polynomial in t, such as 3t^2 - t + 1. The answer is then exact.

The calculator squares the components of T and adds them: the total must be exactly 1.