# Velocity Vector Versus Tangent Line

• lovelylila
In summary, the conversation is about finding the tangent to a curve at a specific point using the position function and relating it to the velocity vector. The slope of the tangent line is equivalent to the velocity at that point. However, the velocity and the tangent line cannot be directly compared as they have different purposes.
lovelylila
I have encountered a problem in my Calculus homework.

I have a position function, r(t)= (t^2)i + (4t)j and in my homework, I am asked to find the tangent to this curve at the point t=3. I did this by finding dy/dx, or 2t/4 @ t=3 is 6/4. However, I am also asked to relate this to the velocity vector for the position function @ t=3, but I don't understand the relationship. Would they share the same slope? Any help is very much appreciated! :-)

lovelylila said:
Would they share the same slope? Any help is very much appreciated! :-)

Well you have the position's function. The tangent line necessarily has a slope of dr/dt, which is equivalent to the velocity so yes.

Oh that makes sense! Thank you very much :-) But they're not the same line...are they?

The velocity has no inherent sense of position, so you can't really compare velocity and a tangent line even at the same time t. A tangent line more or less answers the question "where would you be at some time you knew you were at position r(t) at time t and maintained a constant velocity for all time?", but don't read too far into this.

## 1. What is the difference between a velocity vector and a tangent line?

A velocity vector represents the direction and speed of an object's motion at a specific point in time, while a tangent line is a straight line that touches a curve at a single point, showing the direction of the curve at that point. In other words, a velocity vector represents the instantaneous velocity of an object, while a tangent line represents the instantaneous slope of a curve.

## 2. How are velocity vectors and tangent lines calculated?

Velocity vectors are calculated by dividing the displacement of an object by the time it took to travel that distance. Tangent lines are calculated using the slope formula, which is the change in the y-coordinates divided by the change in the x-coordinates of two points on the curve.

## 3. Why are velocity vectors and tangent lines important in science?

Velocity vectors and tangent lines are important because they help us understand the motion of objects and how they change over time. They are used in various fields of science, such as physics and engineering, to analyze and predict the behavior of objects.

## 4. Can velocity vectors and tangent lines change over time?

Yes, both velocity vectors and tangent lines can change over time. As an object moves, its velocity vector will change, representing the changing direction and speed of its motion. Similarly, as a curve changes shape, the tangent line at any given point will also change.

## 5. How can we use velocity vectors and tangent lines to solve real-world problems?

Velocity vectors and tangent lines can be used to solve real-world problems by helping us understand and predict the behavior of objects in motion. For example, they can be used to calculate the trajectory of a projectile, determine the speed of a moving vehicle, or analyze the motion of a roller coaster. By understanding these concepts, we can make more accurate and informed decisions in various scientific and engineering applications.

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