Exercise 1: Tangent Vector Points in the Direction of Increasing t

about 3 hours ago (Edited)

In this video I show that the derivative or tangent vector of a curve defined by a vector function always points in the direction in which the parameter t is increasing. I show this by analyzing how the curve behaves for positive or negative values of the term h in the definition of derivative for a vector function. For the positive h case, the result is obvious since the secant vector divided by h yields a positive value, hence the tangent vector is in the same direction as the secant vector, which itself is in the direction of increasing t. For the negative h case, the secant vector points in the direction of decreasing t, but dividing by the negative h yields the tangent vector in reverse direction, i.e. in the direction of increasing t!

#math #vectors #calculus #stem #education

10 Exercise 1 Tangent Vector Direction.jpeg

Timestamps

Exercise 1: Tangent vector points in direction of increasing t: 0:00
Solution: Reviewing the definition of the tangent vector or derivative: 1:10
Case where h is greater than zero: 2:48
- Positive he means secant vector and tangent vector point in the same direction, of increasing t: 4:53
Case where h is less than zero: 7:14
- Secant vector points in the direction of decreasing t, but negative h means tangent vector is in reverse direction and towards increasing t: 8:42
Tangent vector is always pointing in the direction of increasing t: 10:23