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Chapter 12: Problem 53

Evaluate \(\langle a, b, a\rangle \times\langle b, a, b\rangle .\) For what nonzero values of \(a\) and \(b\) are the vectors \(\langle a, b, a\rangle\) and \(\langle b, a, b\rangle\) parallel?

Short Answer

Expert verified
Answer: The vectors 鉄╝, b, a鉄� and 鉄╞, a, b鉄� are parallel when a = b or a = -b for nonzero values of a and b.

Step by step solution

01

Finding the cross product of the two vectors

\(\langle a, b, a\rangle \times \langle b, a, b\rangle\) can be calculated by finding the determinant of the matrix formed by unit direction vectors \(\) and the provided vectors. $$ \begin{bmatrix} i & j & k \\ a & b & a \\ b & a & b \end{bmatrix} $$ Now we can find the determinant by expanding across the first row.
02

Calculate the determinant

Next, calculate the determinant of the matrix: $$ cross\_product = i \cdot \det\begin{bmatrix} b & a \\ a & b \end{bmatrix} - j \cdot \det\begin{bmatrix} a & a \\ b & b \end{bmatrix} + k \cdot \det\begin{bmatrix} a & b \\ b & a \end{bmatrix} $$ Now, calculate the three determinants and simplify the resulting cross product vector: $$ cross\_product = i(b^2 - a^2) - j(0) + k(a^2 - b^2) $$ So the cross product of the two given vectors is: $$ \langle b^2 - a^2, 0, a^2 - b^2\rangle $$
03

Determine when the vectors are parallel

Vectors are parallel if their cross product is a zero vector, which means all components are equal to zero. $$ \langle b^2 - a^2, 0, a^2 - b^2\rangle = \langle 0, 0, 0\rangle $$ For the two given nonzero vectors to be parallel, the following conditions must be satisfied: $$ b^2 - a^2 = 0 $$ $$ a^2 - b^2 = 0 $$ Solving these equations simultaneously gives us: $$ a^2 = b^2 $$ Or $$ a = b \space \mathrm{or} \space a = -b $$ So the vectors \(\langle a, b, a \rangle\) and \(\langle b, a, b \rangle\) are parallel when \(a = b\) or \(a = -b\) for nonzero values of \(a\) and \(b\).

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Most popular questions from this chapter

Let \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\). a. Assume that \(\lim _{t \rightarrow a} \mathbf{r}(t)=\mathbf{L}=\left\langle L_{1}, L_{2}, L_{3}\right\rangle,\) which means that $$\begin{aligned} &\lim _{t \rightarrow a}|\mathbf{r}(t)-\mathbf{L}|=0 . \text { Prove that }\\\ &\lim _{t \rightarrow a} f(t)=L_{1}, \quad \lim _{t \rightarrow a} g(t)=L_{2}, \text { and } \lim _{t \rightarrow a} h(t)=L_{3}. \end{aligned}$$ $$\begin{aligned} &\text { b. Assume that } \lim _{t \rightarrow a} f(t)=L_{1}, \lim _{t \rightarrow a} g(t)=L_{2}, \text { and }\\\ &\lim _{t \rightarrow a} h(t)=L_{3} . \text { Prove that } \lim _{t \rightarrow a} \mathbf{r}(t)=\mathbf{L}=\left\langle L_{1}, L_{2}, L_{3},\right\rangle\\\ &\text { which means that } \lim _{t \rightarrow a}|\mathbf{r}(t)-\mathbf{L}|=0. \end{aligned}$$

The definition \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta\) implies that \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|(\text {because}|\cos \theta| \leq 1) .\) This inequality, known as the Cauchy-Schwarz Inequality, holds in any number of dimensions and has many consequences. Use the vectors \(\mathbf{u}=\langle\sqrt{a}, \sqrt{b}\rangle\) and \(\mathbf{v}=\langle\sqrt{b}, \sqrt{a}\rangle\) to show that \(\sqrt{a b} \leq(a+b) / 2,\) where \(a \geq 0\) and \(b \geq 0\).

For the following vectors u and \(\mathbf{v}\) express u as the sum \(\mathbf{u}=\mathbf{p}+\mathbf{n},\) where \(\mathbf{p}\) is parallel to \(\mathbf{v}\) and \(\mathbf{n}\) is orthogonal to \(\mathbf{v}\). \(\mathbf{u}=\langle 4,3\rangle, \mathbf{v}=\langle 1,1\rangle\)

Consider the curve \(\mathbf{r}(t)=(a \cos t+b \sin t) \mathbf{i}+(c \cos t+d \sin t) \mathbf{j}+(e \cos t+f \sin t) \mathbf{k}\) where \(a, b, c, d, e,\) and fare real numbers. It can be shown that this curve lies in a plane. Graph the following curve and describe it. $$\begin{aligned} \mathbf{r}(t)=&\left(\frac{1}{\sqrt{2}} \cos t+\frac{1}{\sqrt{3}} \sin t\right) \mathbf{i}+\left(-\frac{1}{\sqrt{2}} \cos t+\frac{1}{\sqrt{3}} \sin t\right) \mathbf{j} \\ &+\left(\frac{1}{\sqrt{3}} \sin t\right) \mathbf{k} \end{aligned}$$

Trajectory with a sloped landing Assume an object is launched from the origin with an initial speed \(\left|\mathbf{v}_{0}\right|\) at an angle \(\alpha\) to the horizontal, where \(0 < \alpha < \frac{\pi}{2}\) a. Find the time of flight, range, and maximum height (relative to the launch point) of the trajectory if the ground slopes downward at a constant angle of \(\theta\) from the launch site, where \(0 < \theta < \frac{\pi}{2}\) b. Find the time of flight, range, and maximum height of the trajectory if the ground slopes upward at a constant angle of \(\theta\) from the launch site.
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