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Chapter 11: Problem 14

Dot product from the definition Compute \(\mathbf{u} \cdot \mathbf{v}\) if \(\mathbf{u}\) is a unit vector, \(|\mathbf{v}|=2,\) and the angle between them is \(3 \pi / 4\).

Short Answer

Expert verified
Answer: The dot product of the given vectors is \(-\sqrt{2}\).

Step by step solution

01

Recall the dot product definition using the angle between vectors

The dot product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is defined as: \(\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta\), where \(\theta\) is the angle between the two vectors.
02

Plug in the given values

We have \(\mathbf{u}\) as a unit vector, which means \(|\mathbf{u}|=1\). We are also given \(|\mathbf{v}|=2\) and the angle between them \(\theta = \frac{3\pi}{4}\). With these values, we can directly plug them into the dot product definition: \(\mathbf{u} \cdot \mathbf{v} = (1)(2)\cos\left(\frac{3\pi}{4}\right)\).
03

Evaluate \(\cos\frac{3 \pi}{4}\)

To simplify the expression, we first need to compute the cosine of the angle: \(\cos\frac{3 \pi}{4}\). This angle is located in the second quadrant where the cosine of any angle is negative. Therefore, \(\cos\frac{3 \pi}{4} = -\cos\frac{\pi}{4} = -\frac{1}{\sqrt{2}}\).
04

Calculate the dot product

Now that we have evaluated \(\cos\frac{3 \pi}{4}\), we can compute the dot product of the given vectors, by plugging it into the expression we found in step 2: \(\mathbf{u} \cdot \mathbf{v} = (1)(2)\left(-\frac{1}{\sqrt{2}}\right)\). To simplify further, multiply the constants to obtain the final dot product: \(\mathbf{u} \cdot \mathbf{v} = -\frac{2}{\sqrt{2}} = -\sqrt{2}\).

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Most popular questions from this chapter

Suppose water flows in a thin sheet over the \(x y\) -plane with a uniform velocity given by the vector \(\mathbf{v}=\langle 1,2\rangle ;\) this means that at all points of the plane, the velocity of the water has components \(1 \mathrm{m} / \mathrm{s}\) in the \(x\) -direction and \(2 \mathrm{m} / \mathrm{s}\) in the \(y\) -direction (see figure). Let \(C\) be an imaginary unit circle (that does not interfere with the flow). a. Show that at the point \((x, y)\) on the circle \(C\), the outwardpointing unit vector normal to \(C\) is \(\mathbf{n}=\langle x, y\rangle\) b. Show that at the point \((\cos \theta, \sin \theta)\) on the circle \(C,\) the outwardpointing unit vector normal to \(C\) is also \(\mathbf{n}=\langle\cos \theta, \sin \theta\rangle\) c. Find all points on \(C\) at which the velocity is normal to \(C\). d. Find all points on \(C\) at which the velocity is tangential to \(C\). e. At each point on \(C\), find the component of \(\mathbf{v}\) normal to \(C\). Express the answer as a function of \((x, y)\) and as a function of \(\theta\) f. What is the net flow through the circle? That is, does water accumulate inside the circle?

Relationship between \(\mathbf{r}\) and \(\mathbf{r}^{\prime}\) Consider the circle \(\mathbf{r}(t)=\langle a \cos t, a \sin t\rangle,\) for \(0 \leq t \leq 2 \pi\) where \(a\) is a positive real number. Compute \(\mathbf{r}^{\prime}\) and show that it is orthogonal to \(\mathbf{r}\) for all \(t\)

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 \(f\) are real numbers. It can be shown that this curve lies in a plane. Find a general expression for a nonzero vector orthogonal to the plane containing the curve. $$\begin{aligned}\mathbf{r}(t)=&(a \cos t+b \sin t) \mathbf{i}+(c \cos t+d \sin t) \mathbf{j} \\ &+(e \cos t+f \sin t) \mathbf{k},\end{aligned}$$ where \(\langle a, c, e\rangle \times\langle b, d, f\rangle \neq \mathbf{0}\).

Evaluate the following definite integrals. $$\int_{-\pi}^{\pi}(\sin t \mathbf{i}+\cos t \mathbf{j}+2 t \mathbf{k}) d t$$

Prove the following vector properties using components. Then make a sketch to illustrate the property geometrically. Suppose \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) are vectors in the \(x y\) -plane and a and \(c\) are scalars. $$(a+c) \mathbf{v}=a \mathbf{v}+c \mathbf{v}$$
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