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Chapter 12: Problem 24

Define the points \(P(-3,-1)\) \(Q(-1,2), R(1,2), S(3,5), T(4,2),\) and \(U(6,4)\). Sketch \(\overrightarrow{P U}, \overrightarrow{T R},\) and \(\overrightarrow{S Q}\) and the corresponding position vectors.

Short Answer

Expert verified
Answer: The vectors between the given points are: 1. \(\overrightarrow{P U} = 9\hat{i} + 5\hat{j}\) 2. \(\overrightarrow{T R} = -3\hat{i}\) 3. \(\overrightarrow{S Q} = -4\hat{i} - 3\hat{j}\)

Step by step solution

01

Understanding Position Vectors

Position vectors are vectors that define the position of a point in a coordinate system. In a 2D Cartesian plane, a position vector has its tail at the origin and its head at the point it represents. The vector can be represented as \(\binom{x}{y}\) or \(x\hat{i} + y\hat{j}\), where \(x\) and \(y\) are the coordinates of the point.
02

Identify Position Vectors for Each Point

For each given point, we need to find the corresponding position vector: - \(\overrightarrow{OP} = \binom{-3}{-1} = -3\hat{i} - \hat{j}\) - \(\overrightarrow{OQ} = \binom{-1}{2} = -\hat{i} + 2\hat{j}\) - \(\overrightarrow{OR} = \binom{1}{2} = \hat{i} + 2\hat{j}\) - \(\overrightarrow{OS} = \binom{3}{5} = 3\hat{i} + 5\hat{j}\) - \(\overrightarrow{OT} = \binom{4}{2} = 4\hat{i} + 2\hat{j}\) - \(\overrightarrow{OU} = \binom{6}{4} = 6\hat{i} + 4\hat{j}\)
03

Find the Vectors between the Given Points

Now we need to find the vectors between the given points: 1. \(\overrightarrow{P U} = \overrightarrow{OU} - \overrightarrow{OP}\) 2. \(\overrightarrow{T R} = \overrightarrow{OR} - \overrightarrow{OT}\) 3. \(\overrightarrow{S Q} = \overrightarrow{OQ} - \overrightarrow{OS}\) Using the position vectors we found earlier, we can calculate these vectors: 1. \(\overrightarrow{P U} = (6\hat{i} + 4\hat{j}) - (-3\hat{i} - \hat{j}) = 9\hat{i} + 5\hat{j}\) 2. \(\overrightarrow{T R} = (\hat{i} + 2\hat{j}) - (4\hat{i} + 2\hat{j}) = -3\hat{i}\) 3. \(\overrightarrow{S Q} = (-\hat{i} + 2\hat{j}) - (3\hat{i} + 5\hat{j}) = -4\hat{i} - 3\hat{j}\)
04

Sketch the Vectors on a Cartesian Plane

To sketch the vectors, first plot the given points on a Cartesian plane. Then draw arrows representing the vectors from their initial points to their terminal points. Also, draw the position vectors for each point. Here are the steps to sketch the vectors: 1. Plot points P, Q, R, S, T, and U on the Cartesian plane. 2. Draw arrows from point P to point U for vector \(\overrightarrow{P U}\). Label the vector \(9\hat{i} + 5\hat{j}\). 3. Draw arrows from point T to point R for vector \(\overrightarrow{T R}\). Label the vector \(-3\hat{i}\). 4. Draw arrows from point S to point Q for vector \(\overrightarrow{S Q}\). Label the vector \(-4\hat{i} - 3\hat{j}\). 5. Draw position vectors from the origin to each point and label them with their corresponding coordinates. After completing these steps, you should have a clear visual representation of the position vectors and the vectors between the given points.

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Key Concepts

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

Cartesian Plane
The Cartesian plane is a two-dimensional surface on which points are plotted. It is defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point on this plane can be described by an ordered pair of numbers \((x, y)\) known as coordinates.
  • The x-coordinate represents the horizontal position.
  • The y-coordinate represents the vertical position.
To sketch vectors like \(\overrightarrow{P U}\), \(\overrightarrow{T R}\), and \(\overrightarrow{S Q}\), we first plot the points on this plane. Vectors on this plane show the direction and distance from one point to another. Understanding this helps visualize relationships between points and how vectors function in a 2D space.
Vector Arithmetic
Vector arithmetic involves operations like addition, subtraction, and scalar multiplication on vectors. When you subtract one position vector from another, you obtain a vector between two points. For example, to find \(\overrightarrow{P U}\), we subtract \(\overrightarrow{OP}\) from \(\overrightarrow{OU}\):

\[\overrightarrow{P U} = \overrightarrow{OU} - \overrightarrow{OP} = (6\hat{i} + 4\hat{j}) - (-3\hat{i} - \hat{j}) = 9\hat{i} + 5\hat{j}\]
This arithmetic shows:
  • Addition: Combining vectors to find a resultant.
  • Subtraction: Finding the difference between two vectors to show direction and length.
Practicing vector arithmetic is crucial for solving geometry and physics problems involving vectors.
2D Vectors
2D vectors are entities with both magnitude and direction on a flat surface or plane. Each vector has a pair of components, usually represented as \(x\hat{i} + y\hat{j}\), where \(x\) and \(y\) are the vector's projections on the x-axis and y-axis, respectively.
  • Magnitude: Calculated using \(\sqrt{x^2 + y^2}\).
  • Direction: Shown by the angle it makes with the x-axis.
2D vectors help describe motion, forces, and position changes in a plane. Understanding how to operate with these vectors allows for a deeper grasp of spatial relationships and transformations.
Coordinate Geometry
Coordinate geometry combines algebra and geometry to solve problems involving points, lines, and shapes in a plane. It uses coordinates to determine vector relationships and properties. By applying formulas and concepts, one can find distances, midpoints, and slopes.
  • Distance Formula: \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
  • Midpoint Formula: \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\).
In this exercise, you use coordinate geometry to plot and analyze vectors like \(\overrightarrow{P U}\), \(\overrightarrow{T R}\), and \(\overrightarrow{S Q}\). Mastery of coordinate geometry is essential for solving problems in physics, engineering, and various fields of science.

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Most popular questions from this chapter

Determine the equation of the line that is perpendicular to the lines \(\mathbf{r}(t)=\langle-2+3 t, 2 t, 3 t\rangle\) and \(\mathbf{R}(s)=\langle-6+s,-8+2 s,-12+3 s\rangle\) and passes through the point of intersection of the lines \(\mathbf{r}\) and \(\mathbf{R}\).

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. $$\begin{aligned} \mathbf{r}(t)=&\left(\frac{1}{\sqrt{2}} \cos t+\frac{1}{\sqrt{3}} \sin t\right) \mathbf{i}+\left(-\frac{1}{\sqrt{2}} \cos t+\frac{1}{\sqrt{3}} \sin t\right) \mathbf{j} \\ &+\left(\frac{1}{\sqrt{3}} \sin t\right) \mathbf{k} \end{aligned}$$

Let \(\mathbf{u}=\left\langle u_{1}, u_{2}, u_{3}\right\rangle\) \(\mathbf{v}=\left\langle v_{1}, v_{2}, v_{3}\right\rangle\), and \(\mathbf{w}=\) \(\left\langle w_{1}, w_{2}, w_{3}\right\rangle\). Let \(c\) be a scalar. Prove the following vector properties. \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|\)

Consider the ellipse \(\mathbf{r}(t)=\langle a \cos t, b \sin t\rangle\) for \(0 \leq t \leq 2 \pi,\) where \(a\) and \(b\) are real numbers. Let \(\theta\) be the angle between the position vector and the \(x\) -axis. a. Show that \(\tan \theta=(b / a) \tan t\) b. Find \(\theta^{\prime}(t)\) c. Note that the area bounded by the polar curve \(r=f(\theta)\) on the interval \([0, \theta]\) is \(A(\theta)=\frac{1}{2} \int_{0}^{\theta}(f(u))^{2} d u\) Letting \(f(\theta(t))=|\mathbf{r}(\theta(t))|,\) show that \(A^{\prime}(t)=\frac{1}{2} a b\) d. Conclude that as an object moves around the ellipse, it sweeps out equal areas in equal times.

Consider an object moving along the circular trajectory \(\mathbf{r}(t)=\langle A \cos \omega t, A \sin \omega t\rangle,\) where \(A\) and \(\omega\) are constants. a. Over what time interval \([0, T]\) does the object traverse the circle once? b. Find the velocity and speed of the object. Is the velocity constant in either direction or magnitude? Is the speed constant? c. Find the acceleration of the object. d. How are the position and velocity related? How are the position and acceleration related? e. Sketch the position, velocity, and acceleration vectors at four different points on the trajectory with \(A=\omega=1\)
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