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Chapter 12: Problem 12

Express the vector \(\mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle\) in terms of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\)

Short Answer

Expert verified
Question: Express the vector \(\mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle\) in terms of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). Answer: The vector \(\mathbf{v}\) can be expressed in terms of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) as \(\mathbf{v}=v_{1}\mathbf{i}+v_{2}\mathbf{j}\).

Step by step solution

01

Identify the components of the given vector

The given vector \(\mathbf{v}\) is expressed using angle brackets as \(\mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle\). In this form, \(v_{1}\) represents the x-component of the vector, and \(v_{2}\) represents the y-component.
02

Express the vector in terms of the unit vectors i and j

Now that we know the x and y components of the vector \(\mathbf{v}\), we can express it in terms of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). Recall that in a Cartesian coordinate system, the unit vector \(\mathbf{i}\) has an x-component of 1 and a y-component of 0, while the unit vector \(\mathbf{j}\) has an x-component of 0 and a y-component of 1. Using these unit vectors, we can represent the x and y components of \(\mathbf{v}\) as the scalar multiples of \(\mathbf{i}\) and \(\mathbf{j}\) respectively. So we can write the vector \(\mathbf{v}\) as: $$\mathbf{v}=v_{1}\mathbf{i}+v_{2}\mathbf{j}$$

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Most popular questions from this chapter

Evaluate the following limits. $$\lim _{t \rightarrow 2}\left(\frac{t}{t^{2}+1} \mathbf{i}-4 e^{-t} \sin \pi t \mathbf{j}+\frac{1}{\sqrt{4 t+1}} \mathbf{k}\right)$$

Determine whether the following statements are true and give an explanation or counterexample. a. The line \(\mathbf{r}(t)=\langle 3,-1,4\rangle+t\langle 6,-2,8\rangle\) passes through the origin. b. Any two nonparallel lines in \(\mathbb{R}^{3}\) intersect. c. The curve \(\mathbf{r}(t)=\left\langle e^{-t}, \sin t,-\cos t\right\rangle\) approaches a circle as \(t \rightarrow \infty\). d. If \(\mathbf{r}(t)=e^{-t^{2}}\langle 1,1,1\rangle\) then \(\lim _{t \rightarrow \infty} \mathbf{r}(t)=\lim _{t \rightarrow-\infty} \mathbf{r}(t)\).

Evaluate the following limits. $$\lim _{t \rightarrow \infty}\left(e^{-t} \mathbf{i}-\frac{2 t}{t+1} \mathbf{j}+\tan ^{-1} t \mathbf{k}\right)$$

A golfer launches a tee shot down a horizontal fairway and it follows a path given by \(\mathbf{r}(t)=\left\langle a t,(75-0.1 a) t,-5 t^{2}+80 t\right\rangle,\) where \(t \geq 0\) measures time in seconds and \(\mathbf{r}\) has units of feet. The \(y\) -axis points straight down the fairway and the z-axis points vertically upward. The parameter \(a\) is the slice factor that determines how much the shot deviates from a straight path down the fairway. a. With no slice \((a=0),\) sketch and describe the shot. How far does the ball travel horizontally (the distance between the point the ball leaves the ground and the point where it first strikes the ground)? b. With a slice \((a=0.2),\) sketch and describe the shot. How far does the ball travel horizontally? c. How far does the ball travel horizontally with \(a=2.5 ?\)

An object on an inclined plane does not slide provided the component of the object's weight parallel to the plane \(\left|\mathbf{W}_{\text {par }}\right|\) is less than or equal to the magnitude of the opposing frictional force \(\left|\mathbf{F}_{\mathrm{f}}\right|\). The magnitude of the frictional force, in turn, is proportional to the component of the object's weight perpendicular to the plane \(\left|\mathbf{W}_{\text {perp }}\right|\) (see figure). The constant of proportionality is the coefficient of static friction, \(\mu\) a. Suppose a 100 -lb block rests on a plane that is tilted at an angle of \(\theta=20^{\circ}\) to the horizontal. Find \(\left|\mathbf{W}_{\text {parl }}\right|\) and \(\left|\mathbf{W}_{\text {perp }}\right|\) b. The condition for the block not sliding is \(\left|\mathbf{W}_{\mathrm{par}}\right| \leq \mu\left|\mathbf{W}_{\text {perp }}\right| .\) If \(\mu=0.65,\) does the block slide? c. What is the critical angle above which the block slides with \(\mu=0.65 ?\)
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