Something about tangent vector

In summary, the problem is asking for parametric equations that describe specific curves and then finding the tangent vector at each point on those curves. The tangent vector is essentially the velocity vector of the curve.
  • #1
maki1314
2
0
hey there, i got stuck on an question here:
Parameterise the following paths, in the dirction stated, and hence find a tagent vector(in the same dirction) to each point on the paths.
(a)The upper part of the circled centred at (0,0) containing the points (-2,0) and (2,0) going anticlockwise.
(b) the circle centred at (1,2,5) of radius three in the plane z=5 going clockwise (looking down the z axis)

lol,really need help on these two questions. Thx everyone
 
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  • #2
maki1314 said:
hey there, i got stuck on an question here:
Parameterise the following paths, in the dirction stated, and hence find a tagent vector(in the same dirction) to each point on the paths.
(a)The upper part of the circled centred at (0,0) containing the points (-2,0) and (2,0) going anticlockwise.
(b) the circle centred at (1,2,5) of radius three in the plane z=5 going clockwise (looking down the z axis)

lol,really need help on these two questions. Thx everyone

Can you tell us what you have attempted? In particular, do you have a clear understanding of what "tangent vector" means?
 
  • #3
rs1n said:
Can you tell us what you have attempted? In particular, do you have a clear understanding of what "tangent vector" means?

honestly, i attempt another 3 similar questions but really have no idea on that one. all my understanding is that tangent vec. is the trace of a curve at given points...something like that hum? not sure if my understanding is correct...lol
 
  • #4
maki1314 said:
honestly, i attempt another 3 similar questions but really have no idea on that one. all my understanding is that tangent vec. is the trace of a curve at given points...something like that hum? not sure if my understanding is correct...lol

A tangent vector is essentially a vector that is tangent to the graph of your curve. Surely you know what "tangent" and "vector" mean. If you think of your curve as a position function (written in the form of a vector; see below), the tangent vector is equivalent to the velocity vector. The problem requires that you come up with a set of parametric equations x(t) and y(t) that would describe the position of a point on the specified curves. Then, using this position function, determine the tangent vector.

If [itex]\vec{r}(t)[/itex] is the position vector, then

[tex]\vec{r}(t) = x(t) \vec{\mathbf{i}} + y(t) \vec{\mathbf{j}}[/tex]
 

1. What is a tangent vector?

A tangent vector is a mathematical concept used to describe the direction and magnitude of change of a curve or surface at a specific point. It is a vector that is tangent, or perpendicular, to the curve or surface at that point.

2. How is a tangent vector different from a normal vector?

A tangent vector is a vector that lies in the same plane as the curve or surface at a specific point, while a normal vector is a vector that is perpendicular to the curve or surface at that point. Tangent vectors are used to describe the direction of change, while normal vectors are used to describe the orientation of the curve or surface.

3. What is the importance of tangent vectors in calculus?

Tangent vectors are essential in calculus because they are used to calculate rates of change or derivatives. They help us understand the behavior of a curve or surface at a specific point and make it possible to solve problems involving instantaneous change and motion.

4. How do we calculate the magnitude of a tangent vector?

The magnitude of a tangent vector is calculated using the Pythagorean theorem, which states that the length of a vector is equal to the square root of the sum of the squares of its components. In the case of a tangent vector, the components are the partial derivatives of the curve or surface with respect to each variable.

5. Can a tangent vector be negative?

Yes, a tangent vector can be negative. The direction of a tangent vector is determined by the direction of the curve or surface at a specific point, which can be positive or negative depending on the orientation of the curve or surface. However, the magnitude of a tangent vector is always positive.

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