/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 92 \(\vec{A}=3 \hat{i}+4 \hat{j}+2 ... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 92

\(\vec{A}=3 \hat{i}+4 \hat{j}+2 \hat{k}, \vec{B}=6 \hat{i}-\hat{j}+3 \hat{k}\). Find a vector paral- lel to \(\vec{A}\) whose magnitude is equal to that of \(\vec{B}\). (A) \(\sqrt{\frac{46}{29}}(3 \hat{i}+4 \hat{j}+2 \hat{k})\) (B) \(\sqrt{\frac{46}{29}}(6 \hat{i}-\hat{j}+3 \hat{k})\) (C) \(\sqrt{\frac{29}{46}}(3 \hat{i}+4 \hat{j}+2 \hat{k})\) (D) None

Short Answer

Expert verified
The short answer is: \(\vec{C} = \sqrt{\frac{46}{29}}(3\hat{i} + 4\hat{j} + 2\hat{k})\) (Option A).

Step by step solution

01

- Magnitude of \(\vec{A}\) and \(\vec{B}\)

First, let's find the magnitudes of \(\vec{A}\) and \(\vec{B}\) using the formula for the magnitude of a vector: Magnitude \(\vec{A}\) = \(\sqrt{{(3)}^2 + {(4)}^2 + {(2)}^2} = \sqrt{29}\) Magnitude \(\vec{B}\) = \(\sqrt{{(6)}^2 + {(-1)}^2 + {(3)}^2} = \sqrt{46}\)
02

- Determine the scalar

Now, let's find the scalar that should be applied to \(\vec{A}\) to create a new vector that is parallel to \(\vec{A}\) and has the same magnitude as \(\vec{B}\). To do this, we can use the following equation: Scalar = \(\frac{\text{Magnitude of }\vec{B}}{\text{Magnitude of } \vec{A}}\) Scalar = \(\frac{\sqrt{46}}{\sqrt{29}}\)
03

- Multiply the scalar by \(\vec{A}\)

We will now multiply \(\vec{A}\) by the scalar to get the desired vector: \(\vec{C} = \frac{\sqrt{46}}{\sqrt{29}}(3 \hat{i} + 4 \hat{j} + 2 \hat{k})\)
04

- Compare the result with the options

Let's compare our result of \(\vec{C}\) with the given options: (A) \(\sqrt{\frac{46}{29}}(3 \hat{i}+4 \hat{j}+2 \hat{k})\) (B) \(\sqrt{\frac{46}{29}}(6 \hat{i}-\hat{j}+3 \hat{k})\) (C) \(\sqrt{\frac{29}{46}}(3 \hat{i}+4 \hat{j}+2 \hat{k})\) Our answer \( \vec{C} = \frac{\sqrt{46}}{\sqrt{29}}(3 \hat{i} + 4 \hat{j} + 2 \hat{k})\) matches option (A): So, the correct answer is (A).

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Most popular questions from this chapter

The fastest commercial airline service is \(1450 \mathrm{mi} / \mathrm{h}\). Find the speed in \(\mathrm{kmh}^{-1}\) and \(\mathrm{ms}^{-1}\). (A) \(1938 \mathrm{kmh}^{-1}, 618.3 \mathrm{~ms}^{-1}\) (B) \(2030 \mathrm{kmh}^{-1}, 623.1 \mathrm{~ms}^{-1}\) (C) \(2334 \mathrm{kmh}^{-1}, 647.5 \mathrm{~ms}^{-1}\) (D) None

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Given \(\vec{A}=2 \hat{i}+p \hat{j}+q \hat{k}\) and \(\vec{B}=5 \hat{i}+7 \hat{j}+3 \hat{k}\). If \(\vec{A} \| \vec{B}\) then the values of \(p\) and \(q\) are, respectively, (A) \(\frac{14}{5}\) and \(\frac{6}{5}\) (B) \(\frac{14}{3}\) and \(\frac{6}{5}\) (C) \(\frac{6}{5}\) and \(\frac{1}{3}\) (D) \(\frac{3}{4}\) and \(\frac{1}{4}\)
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