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Chapter 17: Problem 1653

\(\underline{a}=(1 / \sqrt{1} 0)(3 \underline{i}+\underline{k}) \underline{b}=(1 / 7)(2 \underline{i}+3 j-6 \underline{k})\) evaluate \((2 \underline{\mathrm{a}}-\underline{\mathrm{b}}) \cdot[(\underline{\mathrm{a}} \times \underline{\mathrm{b}}) \times(\underline{\mathrm{a}}+2 \underline{\mathrm{b}})]\) (a) \(-3\) (b) 5 (c) 3 (d) \(-5\)

Short Answer

Expert verified
The short answer is: \(-3\).

Step by step solution

01

Calculate 2a - b

First, find 2 times vector a and resultant vector b. \(2\vec{a} = (2 / \sqrt{1} 0)(3 \underline{i}+\underline{k}) = 6 \underline{i}+2 \underline{k}\) \(\vec{b} = (1 / 7)(2 \underline{i}+3 j-6 \underline{k}) = \frac{2 \underline{i}}{7}+\frac{3 j}{7}-\frac{6 \underline{k}}{7}\) Now, calculate 2a - b: \(2\vec{a} - \vec{b} = (6\vec{i} + 2\vec{k}) - (\frac{2 \underline{i}}{7}+\frac{3 j}{7}-\frac{6 \underline{k}}{7}) = (\frac{40\vec{i}}{7} + 2\vec{k}) - (\frac{3 j}{7}-\frac{6 \underline{k}}{7})\)
02

Calculate a x b and a + 2b

Now, find the cross product of vector a and vector b: \(\vec{a} \times \vec{b} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ 3 & 0 & 1 \\ \frac{2}{7} & \frac{3}{7} & -\frac{6}{7} \end{vmatrix} = (\frac{3}{7}\vec{j} + \frac{6}{7}\vec{k}) - (-\frac{3}{7}\vec{j}) = \frac{6\vec{i}}{7}+\frac{6\vec{k}}{7}\) Next, find vector a + 2 times vector b: \(\vec{a} + 2\vec{b} = (3 \underline{i}+\underline{k}) + (2 / \sqrt{1} 0)(\frac{2 \underline{i}}{7}+\frac{3 j}{7}-\frac{6 \underline{k}}{7}) = (3 \underline{i}+\underline{k}) + (\frac{4 \underline{i}}{7}+\frac{6 j}{7}-\frac{12 \underline{k}}{7}) = \frac{25\vec{i}}{7}+\frac{6j}{7}-\frac{11\vec{k}}{7}\)
03

Calculate (a x b) x (a + 2b)

Now, find the cross product of the two results obtained in step 2: [((3/7)j + (6/7)k) x ((25/7)i +(6/7)j - (11/7)k)] = (-6i+(19j)+(18k))
04

Calculate the dot product of the results from step 1 and step 3

Lastly, find the dot product of (2a - b) and [(a x b) x (a + 2b)]: (2a - b) · [(a x b) x (a + 2b)] = (\(\frac{40i}{7}-\frac{3j}{7}-\frac{6k}{7}\)) · (-6i + 19j + 18k) \(= -\frac{240}{7} + \frac{57}{7} - \frac{108}{7}\) = \(-\frac{291}{7}\) = -3 So, the answer is (a) -3.

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Most popular questions from this chapter

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