Could I get some help on this vector value function?

In summary, the conversation discusses how to find the time at which a fighter plane, which can only shoot bullets straight ahead, can hit a target located at the origin. The solution involves using the vector parametrization equation and setting the vector r(t) equal to a constant multiple of r'(t). This results in three equations in k and t, from which two can be chosen to solve for t. The significance of setting a k in the equation is to express that one vector is a constant multiple of the other, indicating that they are parallel.
  • #1
dmalwcc89
11
0

Homework Statement



A fighter plane, which can only shoot bullets straight ahead, travels along the path r(t) = <5 - t, 21 - t^2, 3 - (t^3/27)>. Show that there is precisely one time t at which the pilot can hit a target located at the origin.

Homework Equations



I think I am supposed to use a form of the equation for vector parametrization for a tangent line at r(t) = L(t) = r(t) + t[r'(t)].

The Attempt at a Solution



I computed the derivative of r(t): r'(t) = <-1, -2t, -(1/9)t^2>. My book shows examples, but they all have a t value to compute the equation at. I don't quite understand how to find a line that goes through the origin that is tangent to where the fighter is straight ahead, since I don't know that spot either. I graphed it on a 3D parameter graph so I could visualize what is going on, but how do I find these settings?
 
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  • #2
If the pilot can hit the origin the vector r(t) and the vector r'(t) must be parallel and pointed in opposite directions. So r(t)=k*r'(t). Write out that vector equation, giving you three equations in k and t, pick two of them and solve for t.
 
  • #3
Dick said:
If the pilot can hit the origin the vector r(t) and the vector r'(t) must be parallel and pointed in opposite directions. So r(t)=k*r'(t). Write out that vector equation, giving you three equations in k and t, pick two of them and solve for t.

Thank you very much. I was able to solve the problem, but for study purposes, what is the significance of setting a k in there?
 
  • #4
dmalwcc89 said:
Thank you very much. I was able to solve the problem, but for study purposes, what is the significance of setting a k in there?

Just expressing that one vector is a constant multiple of the other. That's what it means to be parallel.
 
  • #5
Dick said:
Just expressing that one vector is a constant multiple of the other. That's what it means to be parallel.

Perfect. Thank you very much for your help.
 

Related to Could I get some help on this vector value function?

1. What is a vector value function?

A vector value function is a mathematical function that maps a set of input values to a corresponding set of output values, where both the input and output values are vectors. It is commonly used in physics and engineering to represent the motion of objects in space and time.

2. How do I know if a function is a vector value function?

A function is considered a vector value function if the output values are vectors. This means that the function has multiple components, such as magnitude and direction, for each input value. You can also check if the function is written in vector notation, such as using bold letters or arrows to represent the vectors.

3. What are the applications of vector value functions?

Vector value functions have various applications in science and engineering, such as in kinematics, electromagnetism, and fluid dynamics. They are used to describe the motion of objects, the behavior of electric and magnetic fields, and the flow of fluids.

4. How do I solve problems involving vector value functions?

To solve problems involving vector value functions, you can use mathematical techniques such as differentiation and integration to find the rate of change or area under the curve. You can also use geometric interpretations, such as vector addition and subtraction, to analyze the behavior of the vectors.

5. Can I graph a vector value function?

Yes, you can graph a vector value function by plotting the vectors on a coordinate system. The x and y values of the vectors represent the input values, while the magnitude and direction of the vectors represent the output values. This allows you to visualize the behavior of the vectors and analyze their changes over time or space.

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