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Chapter 1: Problem 12

Write an expression in rectangular components for the vector that extends from \(\left(x_{1}, y_{1}, z_{1}\right)\) to \(\left(x_{2}, y_{2}, z_{2}\right)\) and determine the magnitude of this vector.

Short Answer

Expert verified
Question: Given two points in 3D space, (3, -1, 4) and (6, 2, 1), find the vector that extends from the first point to the second point and determine its magnitude. Answer: To find the vector extending from (3, -1, 4) to (6, 2, 1), follow the steps outlined in the solution. Calculate the differences in rectangular components: Δx = 6 - 3 = 3 Δy = 2 - (-1) = 3 Δz = 1 - 4 = -3 Combine the differences into a single vector: vec(v) = ⟨3, 3, -3⟩ Determine the magnitude of the vector: |vec(v)| = √(3^2 + 3^2 + (-3)^2) = √27 Therefore, the vector extending from (3, -1, 4) to (6, 2, 1) is vec(v) = ⟨3, 3, -3⟩, and its magnitude is |vec(v)| = √27.

Step by step solution

01

Write down the problem

Given two points in 3D space, \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\), we need to find the vector that extends from the first point to the second point and then find its magnitude.
02

Find the difference in rectangular components

To find the vector extending from \((x_1, y_1, z_1)\) to \((x_2, y_2, z_2)\), we need to find the difference of their respective rectangular components. This can be done as follows: $$ \begin{aligned} \Delta x &= x_2 - x_1 \\ \Delta y &= y_2 - y_1 \\ \Delta z &= z_2 - z_1 \end{aligned} $$
03

Combine the differences into a single vector

Now that we have the difference in rectangular components, we can represent the vector extending from \((x_1, y_1, z_1)\) to \((x_2, y_2, z_2)\) as follows: $$ \vec{v} = \left\langle \Delta x, \Delta y, \Delta z \right\rangle $$
04

Determine the magnitude of the vector

Now that we have the vector \(\vec{v}\) in terms of its rectangular components, we can determine its magnitude using the Euclidean distance formula for a 3D space: $$ \lvert\vec{v}\rvert = \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2} $$ Following these steps, we will be able to find the expression of the vector in rectangular components and determine its magnitude.

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Most popular questions from this chapter

Express the uniform vector field \(\mathbf{F}=5 \mathbf{a}_{x}\) in \((a)\) cylindrical components; (b) spherical components.

If A represents a vector one unit long directed due east, \(\mathbf{B}\) represents a vector three units long directed due north, and \(\mathbf{A}+\mathbf{B}=2 \mathbf{C}-\mathbf{D}\) and \(2 \mathbf{A}-\mathbf{B}=\mathbf{C}+2 \mathbf{D}\), determine the length and direction of \(\mathbf{C}\).

Three vectors extending from the origin are given as \(\mathbf{r}_{1}=(7,3,-2)\), \(\mathbf{r}_{2}=(-2,7,-3)\), and \(\mathbf{r}_{3}=(0,2,3)\). Find \((a)\) a unit vector perpendicular to both \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2} ;(b)\) a unit vector perpendicular to the vectors \(\mathbf{r}_{1}-\mathbf{r}_{2}\) and \(\mathbf{r}_{2}-\mathbf{r}_{3} ;\) (c) the area of the triangle defined by \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2} ;(d)\) the area of the triangle defined by the heads of \(\mathbf{r}_{1}, \mathbf{r}_{2}\), and \(\mathbf{r}_{3}\).

Demonstrate the ambiguity that results when the cross product is used to find the angle between two vectors by finding the angle between \(\mathbf{A}=3 \mathbf{a}_{x}-2 \mathbf{a}_{y}+4 \mathbf{a}_{z}\) and \(\mathbf{B}=2 \mathbf{a}_{x}+\mathbf{a}_{y}-2 \mathbf{a}_{z} .\) Does this ambiguity exist when the dot product is used?

Given the vector field \(\mathbf{E}=4 z y^{2} \cos 2 x \mathbf{a}_{x}+2 z y \sin 2 x \mathbf{a}_{y}+y^{2} \sin 2 x \mathbf{a}_{z}\) for the region \(|x|,|y|\), and \(|z|\) less than 2, find \((a)\) the surfaces on which \(E_{y}=0 ;(b)\) the region in which \(E_{y}=E_{z} ;(c)\) the region in which \(\mathbf{E}=0 .\)
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