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Chapter 12: Problem 2

How many dependent scalar variables does the function \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\) have?

Short Answer

Expert verified
Answer: There are 3 dependent scalar variables in the function \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\).

Step by step solution

01

Identify the scalar components of the vector function

The given function is \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\). This function is a vector in 3D space, and its components are the scalar functions \(f(t)\), \(g(t)\), and \(h(t)\), which represent the x, y, and z coordinates of the vector, respectively.
02

Analyze the dependence of the scalar components on the variable t

Look at the scalar components of the vector function: \(f(t)\), \(g(t)\), and \(h(t)\). Each of these functions is dependent on the variable \(t\). Since \(t\) is the only variable that influences the value of these functions, we can say that each of the components is a dependent scalar variable on \(t\).
03

Count the number of dependent scalar variables

We have identified that the scalar components \(f(t)\), \(g(t)\), and \(h(t)\) are dependent on the variable \(t\). To find the number of dependent scalar variables, we simply need to count the number of scalar components in the function \(\mathbf{r}(t)\). Since there are three scalar components (i.e., \(f(t)\), \(g(t)\), and \(h(t)\)), the function \(\mathbf{r}(t)\) has 3 dependent scalar variables.

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Most popular questions from this chapter

An object moves along an ellipse given by the function \(\mathbf{r}(t)=\langle a \cos t, b \sin t\rangle,\) for \(0 \leq t \leq 2 \pi,\) where \(a > 0\) and \(b > 0\) a. Find the velocity and speed of the object in terms of \(a\) and \(b\) for \(0 \leq t \leq 2 \pi\) b. With \(a=1\) and \(b=6,\) graph the speed function, for \(0 \leq t \leq 2 \pi .\) Mark the points on the trajectory at which the speed is a minimum and a maximum. c. Is it true that the object speeds up along the flattest (straightest) parts of the trajectory and slows down where the curves are sharpest? d. For general \(a\) and \(b\), find the ratio of the maximum speed to the minimum speed on the ellipse (in terms of \(a\) and \(b\) ).

Consider the motion of an object given by the position function $$\mathbf{r}(t)=f(t)\langle a, b, c\rangle+\left(x_{0}, y_{0}, z_{0}\right\rangle, \text { for } t \geq 0$$ where \(a, b, c, x_{0}, y_{0},\) and \(z_{0}\) are constants and \(f\) is a differentiable scalar function, for \(t \geq 0\) a. Explain why this function describes motion along a line. b. Find the velocity function. In general, is the velocity constant in magnitude or direction along the path?

Let \(\mathbf{v}=\langle a, b, c\rangle\) and let \(\alpha, \beta\) and \(\gamma\) be the angles between \(\mathbf{v}\) and the positive \(x\) -axis, the positive \(y\) -axis, and the positive \(z\) -axis, respectively (see figure). a. Prove that \(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1\) b. Find a vector that makes a \(45^{\circ}\) angle with i and \(\mathbf{j}\). What angle does it make with k? c. Find a vector that makes a \(60^{\circ}\) angle with i and \(\mathbf{j}\). What angle does it make with k? d. Is there a vector that makes a \(30^{\circ}\) angle with i and \(\mathbf{j}\) ? Explain. e. Find a vector \(\mathbf{v}\) such that \(\alpha=\beta=\gamma .\) What is the angle?

Note that two lines \(y=m x+b\) and \(y=n x+c\) are orthogonal provided \(m n=-1\) (the slopes are negative reciprocals of each other). Prove that the condition \(m n=-1\) is equivalent to the orthogonality condition \(\mathbf{u} \cdot \mathbf{v}=0\) where \(\mathbf{u}\) points in the direction of one line and \(\mathbf{v}\) points in the direction of the other line.

Consider the parallelogram with adjacent sides \(\mathbf{u}\) and \(\mathbf{v}\). a. Show that the diagonals of the parallelogram are \(\mathbf{u}+\mathbf{v}\) and \(\mathbf{u}-\mathbf{v}\). b. Prove that the diagonals have the same length if and only if \(\mathbf{u} \cdot \mathbf{v}=0\). c. Show that the sum of the squares of the lengths of the diagonals equals the sum of the squares of the lengths of the sides.
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