An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Calculate 3 u = ⟨ 27 , − 6 ⟩ .
Calculate v + w = ⟨ 1 , 4 ⟩ .
Calculate − 5 ( v + w ) = ⟨ − 5 , − 20 ⟩ .
Calculate 3 u − 5 ( v + w ) = ⟨ 27 , − 6 ⟩ + ⟨ − 5 , − 20 ⟩ = ⟨ 22 , − 26 ⟩ . Kelcy's mistake was that she added ⟨ 27 , − 6 ⟩ and ⟨ 5 , − 20 ⟩ instead of ⟨ 27 , − 6 ⟩ and ⟨ − 5 , − 20 ⟩ .
The mistake Kelcy made was that she should have subtracted ⟨ 5 , − 20 ⟩ from ⟨ 27 , − 6 ⟩ , rather than adding. So the answer is: She should have subtracted ( 27 , − 6 ) and ⟨ 5 , − 20 ⟩ , rather than adding. \boxed{\text{She should have subtracted (27,-6) an d \langle 5,-20\rangle , rather than adding.}}
Explanation
Problem Setup and Objective We are given vectors u = ⟨ 9 , − 2 ⟩ , v = ⟨ − 4 , 3 ⟩ , and w = ⟨ 5 , 1 ⟩ . Kelcy calculated 3 u − 5 ( v + w ) incorrectly and got ⟨ 32 , − 26 ⟩ . We need to find her mistake. Let's recalculate the expression step by step.
Calculating 3u First, we calculate 3 u : 3 u = 3 ⟨ 9 , − 2 ⟩ = ⟨ 3 × 9 , 3 × ( − 2 )⟩ = ⟨ 27 , − 6 ⟩
Calculating v + w Next, we calculate v + w : v + w = ⟨ − 4 , 3 ⟩ + ⟨ 5 , 1 ⟩ = ⟨ − 4 + 5 , 3 + 1 ⟩ = ⟨ 1 , 4 ⟩
Calculating -5(v + w) Now, we calculate − 5 ( v + w ) : − 5 ( v + w ) = − 5 ⟨ 1 , 4 ⟩ = ⟨ − 5 × 1 , − 5 × 4 ⟩ = ⟨ − 5 , − 20 ⟩
Calculating 3u - 5(v + w) Finally, we calculate 3 u − 5 ( v + w ) : 3 u − 5 ( v + w ) = ⟨ 27 , − 6 ⟩ + ⟨ − 5 , − 20 ⟩ = ⟨ 27 + ( − 5 ) , − 6 + ( − 20 )⟩ = ⟨ 22 , − 26 ⟩
Identifying Kelcy's Mistake Comparing our result ⟨ 22 , − 26 ⟩ with Kelcy's result ⟨ 32 , − 26 ⟩ , we can see that her mistake occurred in the last step. She incorrectly added ⟨ 27 , − 6 ⟩ and ⟨ 5 , − 20 ⟩ to get ⟨ 32 , − 26 ⟩ . The correct calculation is ⟨ 27 , − 6 ⟩ + ⟨ − 5 , − 20 ⟩ = ⟨ 22 , − 26 ⟩ . Looking at Kelcy's third step ⟨ 27 , − 6 ⟩ − 5 [⟨ − 1 , 4 ⟩] , she incorrectly calculated − 5 ⟨ − 1 , 4 ⟩ as ⟨ 5 , − 20 ⟩ instead of ⟨ − 5 , − 20 ⟩ . The mistake is that she added ⟨ 27 , − 6 ⟩ and ⟨ 5 , − 20 ⟩ instead of adding ⟨ 27 , − 6 ⟩ and ⟨ − 5 , − 20 ⟩ .
Final Answer Therefore, Kelcy should have added ⟨ 27 , − 6 ⟩ and ⟨ − 5 , − 20 ⟩ , but she added ⟨ 27 , − 6 ⟩ and ⟨ 5 , − 20 ⟩ instead.
Examples
Vector operations are used extensively in physics and engineering. For example, when analyzing forces acting on an object, we often represent forces as vectors. If we have multiple forces acting on an object, we can add the vectors to find the net force. Similarly, if we need to scale a force vector (e.g., doubling the force), we multiply the vector by a scalar. These vector operations help us understand and predict the motion of objects under the influence of multiple forces. This is also used in computer graphics to perform transformations on objects, such as rotations, scaling, and translations.
The total charge delivered by the device is 450 C, which corresponds to approximately 2.81 x 10^21 electrons flowing through it in 30 seconds. This is calculated using the formula for current and the charge of a single electron.
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