Vector Analysis: Sketching Hyperbola x^2-y^2=1 with Vector-Valued Function

In summary, the content of the conversation involves solving for the plane curve represented by the vector-valued function r(t)=cosh ti +sinh tj on the interval 0≤t≤5 and showing that the corresponding rectangular equation is x^2-y^2=1. A graphing utility can be used to verify the sketch of the hyperbola. Additionally, the formula for area, A= 1/2(integral)xdy-ydx, is mentioned and it can be used to solve for the area with appropriate substitutions. The formula cosh^2(t)-sinh^2(t)=1 is also mentioned as a useful relation. It appears that there are two separate questions, with the second
  • #1
mikemichiel
7
0
1. Sketch the plane curve represented by the vector-valued function r(t)=cosh ti +sinh tj on the interval 0</(trying to say less then or equal to)t</(also less then or equal to)5. Show that the rectangular equation corresponding to r(t) is the hyperbola x^2-y^2=1. Verify your sketch using a graphing utility to graph the hyperbola.

2. Let P=(coshφ,sinhφ) be the point on the hyperbola corresponding to r(φ) for φ>0. use the formula for the area

A= 1/2(integral)xdy-ydx
I don't know where to begin, can someone please guide me in the right direction.
 
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  • #2
These look a lot like homework problems. Why were they not posted there?

These are labeled "1" and "2". Are they 2 separate questions? "Use the formula for area" to do what?

Presumably you know that [itex]cos^2 x+ sin^2 x= 1[/itex]. Do you know the equivalent formula for cosh and sinh?
 
  • #3
(1)Make x equal to the horizontal component of r(t) and y the vertical component.
Remember that cosh^2(t)-sinh^2(t)=1.


(2) Looks like a line integral, which path are you integrating with? If it is is r(t) from above then substitute x with the horizontal part of r(t) and y with the vertical and relate dx and dy to dt.
 
  • #4
thanks pinu
 

1. What is vector analysis?

Vector analysis is a branch of mathematics that deals with the study of vectors, which are quantities that have both magnitude and direction. It involves the use of vectors to analyze and solve problems in various areas such as physics, engineering, and geometry.

2. What is a hyperbola?

A hyperbola is a type of conic section, which is a curve formed by the intersection of a plane and a cone. It is defined as the set of all points in a plane such that the difference of the distances from two fixed points, called the foci, is constant. In vector analysis, a hyperbola can be represented by a vector-valued function.

3. How do you sketch a hyperbola x^2-y^2=1 with a vector-valued function?

To sketch a hyperbola x^2-y^2=1 with a vector-valued function, we can first rewrite the equation in parametric form as x=sec(t) and y=tan(t). Then, we can use these parametric equations to create a vector-valued function r(t)=[sec(t), tan(t)]. We can plot points on the hyperbola by plugging in different values of t, and then connect these points to create a curve, which will be the sketch of the hyperbola.

4. What is the significance of vector-valued functions in analyzing hyperbolas?

Vector-valued functions are useful in analyzing hyperbolas because they allow us to represent the curve as a single function rather than two separate equations. This makes it easier to find important points on the hyperbola, such as the foci, vertices, and asymptotes. Vector-valued functions also allow us to easily sketch the hyperbola by plugging in different values of the parameter t.

5. Can vector analysis be used to analyze other conic sections?

Yes, vector analysis can be used to analyze other conic sections such as circles, ellipses, and parabolas. For each of these curves, a vector-valued function can be created to represent the curve, making it easier to find key points and sketch the curve. Additionally, vector analysis techniques such as dot products and cross products can be used to find important properties of these curves, such as the radius of a circle or the eccentricity of an ellipse.

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