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Sagot :
Let's analyze the vectors involved in the problem step by step.
### Step 1: Understanding Bruce's Motion
Bruce swims with a speed of 5 meters/second in a direction [tex]$25^{\circ}$[/tex] east of north.
### Step 2: Understanding the Current's Motion
The current moves with a speed of 3.5 meters/second at an angle of [tex]$60^{\circ}$[/tex] west of south.
### Step 3: Calculating Components
1. Bruce's Vector Components:
- East of North:
- This means his vector's angle can be adjusted relative to the standard coordinate system (measured counterclockwise from the positive x-axis).
2. Current's Vector Components:
- West of South:
- This means the current's vector's angle can also be adjusted likewise.
### Step 4: Converting Angles to Standard Position
### Step 5: Breaking Down Vectors into Components
Breaking down the values into horizontal (x) and vertical (y) components:
1. Bruce's Vector (x and y components):
[tex]\[ <2.1130913087034973, 4.531538935183249> \][/tex]
2. Current's Vector (x and y components):
[tex]\[ <-1.7500000000000016, -3.0310889132455348> \][/tex]
### Step 6: Summing Vectors to Get Bruce's Actual Motion
Combining both the vectors will give us the resultant vector of Bruce's actual motion.
[tex]\[ Bruce's \ Actual \ Motion \ = \ <0.3630913087034957, 1.5004500219377146> \][/tex]
### Step 7: Matching Given Choices with Calculated Results
Now let’s match these vectors with the given options:
- Bruce's Vector: [tex]\(<2.11, 4.53>\)[/tex]
- Current's Vector: [tex]\(<-1.75, -3.03>\)[/tex]
- Bruce's Actual Motion: Combining the given numbers results in matches with none of the explicit result terms but we confirmed components as pieces adding together to resultant.
Therefore, the vectors are:
- Bruce's Vector: [tex]\(\mathbf{<2.11, 4.53>}\)[/tex]
- Current's Vector: [tex]\(\mathbf{<-1.75, -3.03>}\)[/tex]
- Bruce's Actual Motion Vector: is the sum hence interpreted on effect of both.
So the correct vectors are:
[tex]\( \begin{array}{ll} Bruce's \ Vector:<2.11, 4.53> \\ Current's \ Vector:<-1.75, -3.03> \end{array} \)[/tex]
### Step 1: Understanding Bruce's Motion
Bruce swims with a speed of 5 meters/second in a direction [tex]$25^{\circ}$[/tex] east of north.
### Step 2: Understanding the Current's Motion
The current moves with a speed of 3.5 meters/second at an angle of [tex]$60^{\circ}$[/tex] west of south.
### Step 3: Calculating Components
1. Bruce's Vector Components:
- East of North:
- This means his vector's angle can be adjusted relative to the standard coordinate system (measured counterclockwise from the positive x-axis).
2. Current's Vector Components:
- West of South:
- This means the current's vector's angle can also be adjusted likewise.
### Step 4: Converting Angles to Standard Position
### Step 5: Breaking Down Vectors into Components
Breaking down the values into horizontal (x) and vertical (y) components:
1. Bruce's Vector (x and y components):
[tex]\[ <2.1130913087034973, 4.531538935183249> \][/tex]
2. Current's Vector (x and y components):
[tex]\[ <-1.7500000000000016, -3.0310889132455348> \][/tex]
### Step 6: Summing Vectors to Get Bruce's Actual Motion
Combining both the vectors will give us the resultant vector of Bruce's actual motion.
[tex]\[ Bruce's \ Actual \ Motion \ = \ <0.3630913087034957, 1.5004500219377146> \][/tex]
### Step 7: Matching Given Choices with Calculated Results
Now let’s match these vectors with the given options:
- Bruce's Vector: [tex]\(<2.11, 4.53>\)[/tex]
- Current's Vector: [tex]\(<-1.75, -3.03>\)[/tex]
- Bruce's Actual Motion: Combining the given numbers results in matches with none of the explicit result terms but we confirmed components as pieces adding together to resultant.
Therefore, the vectors are:
- Bruce's Vector: [tex]\(\mathbf{<2.11, 4.53>}\)[/tex]
- Current's Vector: [tex]\(\mathbf{<-1.75, -3.03>}\)[/tex]
- Bruce's Actual Motion Vector: is the sum hence interpreted on effect of both.
So the correct vectors are:
[tex]\( \begin{array}{ll} Bruce's \ Vector:<2.11, 4.53> \\ Current's \ Vector:<-1.75, -3.03> \end{array} \)[/tex]
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