An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Find the position vector of point D using the property that opposite sides of a parallelogram are equal: d = a − b + c = − i + 14 j + 4 k .
Calculate the vector A B = b − a = 3 i − 4 j + 5 k and its magnitude ∣ A B ∣ = 5 2 .
Determine the scalar λ such that ∣ A X ∣ = ∣ λ A B ∣ = 15 , which gives λ = 2 3 .
Find the position vector of point X using x = a + λ A B = ( 1 + 2 9 2 ) i + ( 7 − 6 2 ) j + ( − 2 + 2 15 2 ) k .
The position vector of D is − i + 14 j + 4 k and the position vector of X is approximately 7.36 i − 1.49 j + 8.61 k .
Explanation
Problem Analysis and Given Data We are given the position vectors of points A , B , and C as a = i + 7 j − 2 k , b = 4 i + 3 j + 3 k , and c = 2 i + 10 j + 9 k respectively. We also know that A BC D is a parallelogram, and A X has the same direction as A B and ∣ A X ∣ = 15 . We need to find the position vector of point D and the position vector of point X .
Finding the position vector of D (a) Since A BC D is a parallelogram, we know that A B = D C . This means that b − a = c − d , where d is the position vector of point D . We can rearrange this equation to solve for d : d = a − b + c Now, we substitute the given position vectors into the equation: d = ( i + 7 j − 2 k ) − ( 4 i + 3 j + 3 k ) + ( 2 i + 10 j + 9 k ) d = ( 1 − 4 + 2 ) i + ( 7 − 3 + 10 ) j + ( − 2 − 3 + 9 ) k d = − 1 i + 14 j + 4 k So, the position vector of point D is − i + 14 j + 4 k .
Finding the position vector of X (b) Since A X has the same direction as A B , we can write A X = λ A B for some scalar λ . We are given that ∣ A X ∣ = 15 .
First, we need to find the vector A B :
A B = b − a = ( 4 i + 3 j + 3 k ) − ( i + 7 j − 2 k ) = 3 i − 4 j + 5 k Next, we find the magnitude of A B :
∣ A B ∣ = 3 2 + ( − 4 ) 2 + 5 2 = 9 + 16 + 25 = 50 = 5 2 Since ∣ A X ∣ = ∣ λ A B ∣ = ∣ λ ∣∣ A B ∣ = 15 , we have: ∣ λ ∣ ( 5 2 ) = 15 ∣ λ ∣ = 5 2 15 = 2 3 Since A X has the same direction as A B , λ is positive, so λ = 2 3 .
Now, we can find the position vector of X :
x = a + A X = a + λ A B = ( i + 7 j − 2 k ) + 2 3 ( 3 i − 4 j + 5 k ) x = i + 7 j − 2 k + 2 9 i − 2 12 j + 2 15 k x = ( 1 + 2 9 ) i + ( 7 − 2 12 ) j + ( − 2 + 2 15 ) k x = ( 1 + 2 9 2 ) i + ( 7 − 6 2 ) j + ( − 2 + 2 15 2 ) k x ≈ 7.36 i − 1.49 j + 8.61 k
Examples
Understanding vectors and parallelograms is crucial in many real-world applications. For example, in computer graphics, vectors are used to represent the position and orientation of objects in 3D space. Parallelograms can be used to model the deformation of materials under stress. Also, in physics, vectors are used to represent forces and velocities, and the properties of parallelograms are used to analyze the equilibrium of forces. For instance, if you're designing a bridge, you need to understand how forces are distributed and balanced, and vector calculations with parallelograms can help you ensure the bridge's stability.
The total charge flowing through the device is 450 coulombs over the 30 seconds. This results in approximately 2.81 x 10^21 electrons flowing through the device. The calculation involves the relationship between current, time, and charge, along with the charge of a single electron.
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