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Chapter 2: Problem 271

$\mathrm{A}^{\boldsymbol{\longrightarrow}}=2 \mathrm{i} \wedge+2 \mathrm{j} \wedge-\mathrm{k} \wedge\( and \)\mathrm{B}^{\rightarrow}=2 \mathrm{i} \wedge-\mathrm{j} \wedge-2 \mathrm{k} \wedge\( Find \)3 \mathrm{~A}^{\rightarrow}-2 \mathrm{~B}^{\rightarrow}$ (A) \(2 \mathrm{i} \wedge+7 \mathrm{j} \wedge+\mathrm{k} \wedge\) (B) \(2 \mathrm{i} \wedge+8 \mathrm{j} \wedge-\mathrm{k} \wedge\) (C) \(2 \mathrm{i} \wedge+8 \mathrm{j} \wedge+\mathrm{k} \wedge\) (D) \(\mathrm{i} \wedge+7 \mathrm{j} \wedge+\mathrm{k} \wedge\)

Short Answer

Expert verified
The short answer is: \(3\mathrm{A} - 2\mathrm{B} = 2\mathrm{i} + 8\mathrm{j} + \mathrm{k}\)

Step by step solution

01

Multiply Vector A by 3

To multiply vector A by 3, we need to multiply each of its components by 3: \(3\mathrm{A} = 3(2\mathrm{i} + 2\mathrm{j} - \mathrm{k}) = 6\mathrm{i} + 6\mathrm{j} - 3\mathrm{k}\)
02

Multiply Vector B by -2

To multiply vector B by -2, we need to multiply each of its components by -2: \(-2\mathrm{B} = -2(2\mathrm{i} - \mathrm{j} -2\mathrm{k}) = -4\mathrm{i} + 2\mathrm{j} + 4\mathrm{k}\)
03

Add the results of Step 1 and Step 2

Now, we'll add the resulting vectors from Steps 1 and 2: \(3\mathrm{A} - 2\mathrm{B} = (6\mathrm{i} + 6\mathrm{j} - 3\mathrm{k}) + (-4\mathrm{i} + 2\mathrm{j} + 4\mathrm{k})\)
04

Write the final result

Now, we combine the corresponding components in the expression above: \(3\mathrm{A} - 2\mathrm{B} = (6\mathrm{i} - 4\mathrm{i}) + (6\mathrm{j} + 2\mathrm{j}) + (-3\mathrm{k} + 4\mathrm{k}) = 2\mathrm{i} + 8\mathrm{j} + \mathrm{k}\) So, the final result for \(3\mathrm{A} - 2\mathrm{B}\) is \(2\mathrm{i} + 8\mathrm{j} + \mathrm{k}\), and the correct answer is (C).

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