Tangent and normal vectors describe how a particle moves along a curve and how its direction changes. The tangent vector points in the direction of motion, while the normal vector points toward the direction the path is bending. These vectors are important in calculus, physics, and engineering because they connect geometry to velocity, acceleration, and curvature.
They let us describe motion along curved paths without needing only horizontal and vertical components.
Key Facts
- Velocity is the derivative of position: v(t) = r'(t).
- Speed is the magnitude of velocity: speed = |r'(t)|.
- The unit tangent vector is T(t) = r'(t)/|r'(t)| when r'(t) is not zero.
- The unit normal vector points in the direction that T changes: N(t) = T'(t)/|T'(t)| when T'(t) is not zero.
- Acceleration splits into tangential and normal parts: a = a_T T + a_N N.
- For motion along a curve, a_T = d|v|/dt and a_N = kappa |v|^2, where kappa is curvature.
Vocabulary
- Tangent vector
- A tangent vector points in the instantaneous direction of a curve or a particle moving along the curve.
- Unit tangent vector
- The unit tangent vector is a tangent vector scaled to have length 1.
- Normal vector
- A normal vector points perpendicular to the tangent direction and toward the way the curve is turning.
- Curvature
- Curvature measures how quickly a curve changes direction per unit distance traveled.
- TNB frame
- The TNB frame is a moving coordinate frame made of the tangent T, normal N, and binormal B vectors along a space curve.
Common Mistakes to Avoid
- Using r(t) itself as the tangent vector is wrong because the tangent direction comes from the derivative r'(t), not the position vector from the origin.
- Forgetting to normalize T is wrong because the unit tangent vector must have length 1, so T = r'(t)/|r'(t)|.
- Assuming acceleration always points tangent to the path is wrong because curved motion has a normal acceleration component that points toward the bend.
- Treating the normal vector as any perpendicular vector is wrong because the principal normal points in the direction T is changing, not just in either perpendicular direction.
Practice Questions
- 1 For r(t) = <3t, 4t>, find v(t), the speed, and the unit tangent vector T(t).
- 2 For r(t) = <cos t, sin t>, find T(t) and N(t) at t = pi/2.
- 3 A car moves around a curve at constant speed. Explain why its tangential acceleration is zero but its normal acceleration is not zero.