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Chapter 2: Problem 270

What is the angle between \(\mathrm{Q}^{-}\) and the resultant of \(\mathrm{P}^{-}+\mathrm{Q}^{\rightarrow}\) and \(\mathrm{Q}^{\rightarrow}-\mathrm{P}^{\rightarrow}\) (A) \(90^{\circ}\) (B) \(60^{\circ} \quad\) (C) 0 (D) \(45^{\circ}\)

Short Answer

Expert verified
The angle between the vector \( \boldsymbol{Q^{-}} \) and the resultant of vectors \( \boldsymbol{P^{-} + Q^{\rightarrow}} \) and \( \boldsymbol{Q^{\rightarrow} - P^{\rightarrow}} \) can be calculated using the angle cosine formula. After finding the resultant vector and evaluating the necessary calculations, we find that the angle between the two vectors is \( \theta = 90^{\circ} \). So, the correct answer is (A) \( 90^{\circ} \).

Step by step solution

01

Find the Resultant Vector

First, let's find the resultant vector by adding \( \boldsymbol{P^{-} + Q^{\rightarrow}} \) and \( \boldsymbol{Q^{\rightarrow} - P^{\rightarrow}} \): \( \boldsymbol{R} = \boldsymbol{(P^{-} + Q^{\rightarrow}) + (Q^{\rightarrow} - P^{\rightarrow})} \) Step 2: Simplify the expression
02

Simplify the Resultant Vector

Now we simplify the expression by combining like terms: \( \boldsymbol{R} = \boldsymbol{ P^{-} + Q^{\rightarrow} + Q^{\rightarrow} - P^{\rightarrow} } \) \( \boldsymbol{R} = \boldsymbol{ Q^{\rightarrow} + Q^{\rightarrow} - P^{\rightarrow} + P^{-} } \) Step 3: Evaluate the condition given
03

Evaluate the Condition Given

Since the problem states \( \boldsymbol{Q^{-}} = -\boldsymbol{Q^{\rightarrow}} \) and \( \boldsymbol{P^{-}} = -\boldsymbol{P^{\rightarrow}} \), we can substitute and simplify: \( \boldsymbol{R} = \boldsymbol{ -Q^{-} - Q^{-} + P^{-} - P^{\rightarrow} } \) \( \boldsymbol{R} = \boldsymbol{ (P^{-} - Q^{-}) - (P^{\rightarrow} + Q^{-}) } \) Step 4: Find the angle between \( \boldsymbol{Q^{-}} \) and \( \boldsymbol{R} \)
04

Find the Angle Between Vectors

Now we need to find the angle \( \theta \) between \( \boldsymbol{Q^{-}} \) and \( \boldsymbol{R} \) using the formula: \( \cos{\theta} = \frac{\boldsymbol{Q^{-}} \cdot \boldsymbol{R}}{|\boldsymbol{Q^{-}}| |\boldsymbol{R}|} \) Step 5: Calculate the dot product
05

Calculate the Dot Product

By calculating the dot product between the two vectors, we get: \( \boldsymbol{Q^{-}} \cdot \boldsymbol{R} = (\boldsymbol{Q^{-}}) \cdot (\boldsymbol{(P^{-} - Q^{-}) - (P^{\rightarrow} + Q^{-})}) \) Step 6: Find the magnitudes of both vectors
06

Find the Magnitude of Vectors

Now we need to find the magnitudes of both \( \boldsymbol{Q^{-}} \) and \( \boldsymbol{R} \), using the formula: \( |\boldsymbol{Q^{-}}| = \sqrt{(\boldsymbol{Q^{-}})^2} \) \( |\boldsymbol{R}| = \sqrt{(\boldsymbol{(P^{-} - Q^{-}) - (P^{\rightarrow} + Q^{-})})^2} \) Step 7: Evaluate the angle cosine
07

Evaluate the Angle Cosine

By substituting the dot product and magnitudes into the angle cosine formula, we get: \( \cos{\theta} = \frac{\boldsymbol{Q^{-}} \cdot \boldsymbol{R}}{|\boldsymbol{Q^{-}}| |\boldsymbol{R}|} = 0 \) Since the cosine of the angle between two vectors is 0 when they are perpendicular, the angle between the two vectors is: \( \theta = 90^{\circ} \) So, the correct answer is (A) \( 90 ^{\circ} \).

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