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Chapter 1: Problem 34

If \(\vec{a}\) and \(\vec{b}\) are two vectors then the value of \((\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})\) is (A) \(2(\vec{b} \times \vec{a})\) (B) \(-2(\vec{b} \times \vec{a})\) (C) \(\vec{b} \times \vec{a}\) (D) \(\vec{a} \times \vec{b}\)

Short Answer

Expert verified
The short answer to the question is: (B) \(-2(\vec{b} \times \vec{a})\)

Step by step solution

01

Apply the distributive property

First, we apply the distributive property on the given expression: \((\vec{a}+\vec{b}) \times(\vec{a}-\vec{b}) = \vec{a} \times\vec{a} - \vec{a}\times\vec{b} + \vec{b}\times\vec{a} - \vec{b}\times\vec{b}\) Step 2: Use cross-product properties to simplify the expression
02

Simplify using cross-product properties

Now we can use the property for the cross product of a vector with itself, which is equal to the zero vector: \(\vec{a}\times\vec{a} = \vec{0}\) and \(\vec{b}\times\vec{b} = \vec{0}\) So now, we have: \(\vec{0} - \vec{a}\times\vec{b} + \vec{b}\times\vec{a} - \vec{0}\) But we know that: \(\vec{b}\times\vec{a}\) = \(-\vec{a}\times\vec{b}\) So, our expression becomes: \(\vec{0} - \vec{a}\times\vec{b} -\vec{a}\times\vec{b} - \vec{0}\) There is a common factor between the two terms, so we can rewrite the expression as: \(-2(\vec{a}\times\vec{b})\) Step 3: Identify which choice the expression matches
03

Match expression with choices

Now that we have simplified our expression, we can look at the given choices and see which one matches the expression. We found that our simplified expression is \(-2(\vec{a}\times\vec{b})\), and by comparing it to the available options, we find that it matches option B. Therefore, the correct answer is: (B) \(-2(\vec{b} \times \vec{a})\)

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Most popular questions from this chapter

In a two-dimensional motion of a particle, the particle moves from point \(A\), with position vector \(\vec{r}_{1}\), to point \(B\), with position vector \(\vec{r}_{2}\). If the magnitudes of these vectors are, respectively, \(r_{1}=3\) and \(r_{2}=4\) and the angles they make with the \(x\)-axis are \(\theta_{1}=75^{\circ}\) and \(\theta_{2}=15^{\circ}\), respectively, then find the magnitude of the displacement vector. (A) 15 (B) \(\sqrt{13}\) (C) 17 (D) \(\sqrt{15}\)

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