What is the definition of the unit tangent vector T in terms of direction angles?

Conditions

• The curve is smooth at the point of interest.
• $\alpha$ is the angle between the tangent and the positive x-axis.
• $\beta$ is the angle between the tangent and the positive y-axis.
• $\vec{T}$ is a unit vector (magnitude 1).

Reasoning, step by step

1. Identify the tangent line to the curve at a specific point.
2. Measure the angle $\alpha$ between this tangent and the positive x-axis.
3. Measure the angle $\beta$ between this tangent and the positive y-axis.
4. Compute the cosine of these angles: $\cos \alpha$ and $\cos \beta$.
5. Construct the vector $\vec{T} = (\cos \alpha, \cos \beta)$.
6. Verify that $|\vec{T}| = \sqrt{\cos^2 \alpha + \cos^2 \beta} = 1$ (in 2D, assuming appropriate angle definitions).
7. Use this vector to define the direction of the differential elements $dx$ and $dy$.

Example

The script states: 'An orange arrow represents the unit tangent vector at a specific point on the curve. Its components are defined by the angles α and β made with the x and y axes respectively, so $T = (\cos α, \cos β)$.'

Common misconceptions

• Confusing the unit tangent vector with the velocity vector (which may not have magnitude 1).
• Assuming $\alpha$ and $\beta$ are independent; they are related by the geometry of the curve.
• Thinking the direction angles are measured from the negative axes by default.

Watch the explanation

Explore next

• How does the unit tangent vector relate to the derivative of the position vector?
• Why is it important that T is a unit vector for the conversion formula?
• What happens to the direction angles if the curve is traversed in reverse?