Given $|r| = 6$ at an angle of $30^{\circ}$ and $|s| = 11$ at an angle of $225^{\circ}$, which expression represents $|r - s|$?
A. $\sqrt{6^2+11^2-2(6)(11) \cos \left(15^{\circ}\right)}$
B. $\sqrt{6^2+11^2-2(6)(11) \cos \left(45^{\circ}\right)}$
C. $\sqrt{6^2+11^2-2(6)(11) \cos \left(165^{\circ}\right)}$
D. $\sqrt{6^2+11^2-2(6)(11) \cos \left(255^{\circ}\right)}$
Answer (2)
Use the law of cosines to express the magnitude of the difference between two vectors: ∣ r − s ∣ = ∣ r ∣ 2 + ∣ s ∣ 2 − 2∣ r ∣∣ s ∣ cos ( θ ) .
Calculate the angle between the vectors: θ = ∣22 5 ∘ − 3 0 ∘ ∣ = 19 5 ∘ . Recognize that cos ( 19 5 ∘ ) = cos ( 16 5 ∘ ) .
Substitute the given magnitudes and the angle into the law of cosines formula.
The expression representing ∣ r − s ∣ is: 6 2 + 1 1 2 − 2 ( 6 ) ( 11 ) cos ( 16 5 ∘ ) .
Explanation
Problem Analysis We are given two vectors, r and s , with magnitudes ∣ r ∣ = 6 and ∣ s ∣ = 11 , and angles 3 0 ∘ and 22 5 ∘ respectively. We want to find an expression for the magnitude of the difference between these vectors, ∣ r − s ∣ .
Law of Cosines The magnitude of the difference of two vectors can be found using the law of cosines. If we consider the vectors r and s as sides of a triangle, then r − s is the third side, and the angle between r and s is θ . The law of cosines states that: ∣ r − s ∣ 2 = ∣ r ∣ 2 + ∣ s ∣ 2 − 2∣ r ∣∣ s ∣ cos ( θ ) Therefore, ∣ r − s ∣ = ∣ r ∣ 2 + ∣ s ∣ 2 − 2∣ r ∣∣ s ∣ cos ( θ )
Finding the Angle First, we need to find the angle between the two vectors. The angle between r and s is the absolute difference between their angles: θ = ∣22 5 ∘ − 3 0 ∘ ∣ = ∣19 5 ∘ ∣ = 19 5 ∘ Since the cosine function is periodic with a period of 36 0 ∘ , we can also express this angle as: θ = 36 0 ∘ − 19 5 ∘ = 16 5 ∘ Also, cos ( x ) = cos ( − x ) , so cos ( 19 5 ∘ ) = cos ( − 19 5 ∘ ) .
Substituting Values Now, we substitute the given values into the formula: ∣ r − s ∣ = 6 2 + 1 1 2 − 2 ( 6 ) ( 11 ) cos ( 19 5 ∘ ) Since cos ( 19 5 ∘ ) = cos ( 16 5 ∘ ) , we can write: ∣ r − s ∣ = 6 2 + 1 1 2 − 2 ( 6 ) ( 11 ) cos ( 16 5 ∘ ) This matches one of the given options.
Final Answer Therefore, the expression that represents ∣ r − s ∣ is: 6 2 + 1 1 2 − 2 ( 6 ) ( 11 ) cos ( 16 5 ∘ )
Examples
Understanding vector subtraction is crucial in physics, especially when analyzing forces acting on an object. For instance, imagine two people pulling a box with different forces (vectors). By finding the magnitude of the difference between these force vectors, you can determine the net force acting on the box, which dictates its movement. This principle is fundamental in mechanics and engineering for designing structures and predicting their behavior under various loads.
The expression representing ∣ r − s ∣ is derived using the law of cosines. By calculating the angle between the two vectors and substituting the values, the final expression is 6 2 + 1 1 2 − 2 ( 6 ) ( 11 ) cos ( 16 5 ∘ ) . Therefore, the correct answer is option C.
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