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Chapter 21: Problem 23

Does the plane \(\vec{r}(s, t)=(2+s) \vec{i}+(3+s+t) \vec{j}+4 t \vec{k}\) contain the following points? (a) \(\quad(4,8,12)\) (b) \(\quad(1,2,3)\)

Short Answer

Expert verified
(a) Yes, (b) No.

Step by step solution

01

Express the plane equation

The plane is given by the vector function \( \vec{r}(s, t) = (2+s) \vec{i} + (3+s+t) \vec{j} + 4t \vec{k} \). This means any point on the plane can be expressed as the vector **\((x,y,z)\)**, which translates to:\[\begin{align*}x &= 2 + s, \y &= 3 + s + t, \z &= 4t.\end{align*}\]
02

Verify if point (4, 8, 12) lies on the plane

To check if the point \((4, 8, 12)\) lies on the plane, substitute \(x=4\), \(y=8\), and \(z=12\) into the plane equations:- From \(x = 2 + s\): \(4 = 2 + s \Rightarrow s = 2\).- Substitute \(s = 2\) into \(z = 4t\): \(12 = 4t\Rightarrow t = 3\).- Now, substitute \(s = 2\) and \(t = 3\) into \(y = 3 + s + t\): \(8 = 3 + 2 + 3 = 8\), which matches.Thus, the point \((4, 8, 12)\) lies on the plane.
03

Verify if point (1, 2, 3) lies on the plane

To check if the point \((1, 2, 3)\) lies on the plane, substitute \(x=1\), \(y=2\), and \(z=3\) into the plane equations:- From \(x = 2 + s\): \(1 = 2 + s \Rightarrow s = -1\).- Substitute \(s = -1\) into \(z = 4t\): \(3 = 4t\Rightarrow t = \frac{3}{4}\).- Now, substitute \(s = -1\) and \(t = \frac{3}{4}\) into \(y = 3 + s + t\): \(2 = 3 + (-1) + \frac{3}{4} = 2.75 \), which does not match.Therefore, the point \((1, 2, 3)\) does not lie on the plane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Equations
A vector equation is a way to represent geometric figures, like lines and planes, in terms of vectors. It involves expressing points in relation to other points using vectors. For instance, the vector equation for a plane might take the form \( \vec{r}(s, t) = (2+s) \vec{i} + (3+s+t) \vec{j} + 4t \vec{k} \). This expression shows how you can move from the origin, or another point of reference, by applying certain steps, denoted by parameters like \(s\) and \(t\), to find any point on the plane.
  • Parameters: These are variables that help determine points on the line or plane.
  • Vectors: Components that define direction and magnitude.
  • Standard Form: Often involves known fixed values that set initial positions.
Understanding vector equations can greatly help in visualizing and solving spatial problems.
Plane Equations
Plane equations come in different forms, but vector functions are particularly useful in multivariable calculus. Specifically, they help us understand how surfaces stretch through space. A plane equation like \( \vec{r}(s, t) = (2+s) \vec{i} + (3+s+t) \vec{j} + 4t \vec{k} \) characterizes the entire plane.
  • Coordinate System: Breaks down the space into \(x\), \(y\), and \(z\) components.
  • Parameterization: Uses parameters \(s\) and \(t\) to define every point.
  • Infinite Plane: When solving, you essentially map an infinite set of solutions.
This method helps us understand how different points lay on or off the plane by evaluating their coordinates in the given equation.
Point in Space
Identifying a point in space, such as \(4, 8, 12\), involves determining whether it exists on a particular plane or line. To check if a point is on a plane, we use the plane's equation and see if substituting the point's coordinates respects the equation.
  • X-Coordinate: Aligns with the equation for \(x\).
  • Y-Coordinate: Completes the equation for \(y\) with given parameters.
  • Z-Coordinate: Finishes out the set for \(z\).
If the substituted values satisfy the plane equation, then the point exists on the plane. Otherwise, it lies elsewhere in space.
Vector Function
A vector function is an equation that takes several variables (usually parameters) and maps them to a vector. They are highly effective at describing curves, surfaces, and other geometrical structures in space. The vector function for a plane, such as \( \vec{r}(s, t) = (2+s) \vec{i} + (3+s+t) \vec{j} + 4t \vec{k} \), describes entire regions of space.
  • Input Variables: Determine the scope L of the function.
  • Output: A vector showing direction and magnitude.
  • Applications: Solving geometry problems and physical phenomena.
In multivariable calculus, vector functions are core to analyzing dynamic systems and forms.

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Most popular questions from this chapter

Describe the families of parameter curves with \(s=s_{0}\) and \(t=t_{0}\) for the parameterized surface. $$\begin{aligned}&x=s, \quad y=\cos t, \quad z=\sin t \text { for }-\infty

Two independent random numbers \(x\) and \(y\) from a normal distribution with mean 0 and standard deviation \(\sigma\) have joint density function \(p(x, y)=\) \(\left(1 /\left(2 \pi \sigma^{2}\right) e^{-\left(x^{2}+y^{2}\right) /\left(2 \sigma^{2}\right)} . \text { The average } z=(x+y) / 2\) has \right. a one-variable probability density function of its own. (a) Give a double integral expression for \(F(t),\) the probability that \(z \leq t\) (b) Give a single integral expression for \(F(t) .\) To do this, make the change of coordinates: \(u=(x+y) / 2\) \(v=(x-y) / 2\) and then do the integral on \(d v .\) Use the fact that \(\int_{-\infty}^{\infty} e^{-x^{2} / a^{2}} d x=a \sqrt{\pi}\) (c) Find the probability density function \(F^{\prime}(t)\) of \(z\) (d) What is the name of the distribution of z?

Match the parameterizations (I)-(IV) with the surfaces (a)-(d). In all cases \(0 \leq s \leq \pi / 2,0 \leq t \leq \pi / 2 .\) Note that only part of the surface may be described by the given parameterization. (a) Cylinder (b) Plane (c) Sphere (d) Cone I. \(x=\cos s, \quad y=\sin t, \quad z=\cos s+\sin t\) II. \(x=\cos s, \quad y=\sin s, \quad z=\cos t\) III. \(\quad x=\sin s \cos t, \quad y=\sin s \sin t, \quad z=\cos s\) IV. \(x=\cos s, \quad y=\sin t, \quad z=\sqrt{\cos ^{2} s+\sin ^{2} t}\)

Give parametric equations for the plane through the point with position vector \(\vec{r}_{0}\) and containing the vectors \(\vec{v}_{1}\) and \(\vec{v}_{2}\). $$\vec{r}_{0}=\vec{i}+\vec{j}, \vec{v}_{1}=\vec{j}+\vec{k}, \vec{v}_{2}=\vec{i}+\vec{k}$$

Describe the families of parameter curves with \(s=s_{0}\) and \(t=t_{0}\) for the parameterized surface. $$\begin{aligned}&x=s \quad y=t, \quad z=s^{2}+t^{2} \text { for }-\infty
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