Vector & Scalar Homework: Base Camp to Lake B

In summary, the problem involves a plane flying from base camp to Lake A at an angle of 20° north of east, then to Lake B at an angle of 30° west of north. The distance and direction from Lake B to the base camp can be determined graphically by calculating the components of vector A and B, and adding them together to get R. The return route, R, is in the third quadrant and has a distance of 310km and a direction of 57.2° south of west.
  • #1
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Homework Statement


A plane flies from base camp to Lake A, 280 km away inthe direction 20.0°north of east. After dropping off sup-plies, it flies to Lake B, which is 190 km at 30.0°west of north from Lake A. Graphically determine the distance and direction from Lake B to the base camp.

Homework Equations





The Attempt at a Solution


I've drawn a line from base camp to lake A with an angle of 20°, another line from lake A to lake with an angle of 30°.
I have calculated the components of vector A, I already know the [tex]|A|=280km, |B|=190km[/tex], so:
[tex]A_{x}=Acos(\frac{pi}{9})=263km[/tex],
[tex]A_{y}=Asin(\frac{pi}{9})=95.8km[/tex],
[tex]B_{x}=Bsin(\frac{-pi}{6})=-95km[/tex],
[tex]B_{y}=Bcos(\frac{-pi}{6})=165km[/tex];
[tex]R^→=A^→+B^→[/tex];
[tex]|R|=sqrt((168^2)+(261^2))=310km[/tex],
[tex]cosσ=\frac{168}{310}→σ=57.2°[/tex],
[tex]sinσ=\frac{261}{310}→σ=57.4°[/tex];
I get the exact result because:
[tex]B_{x}=Bsin(\frac{-pi}{6})=-95km[/tex],
[tex]B_{y}=Bcos(\frac{-pi}{6})=165km[/tex];
i don't know why,
In which quadrant are the vector B and R?
 
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  • #2
"30 degrees West of North" is in the second quadrant (the x-component is negative ("West") and the y component is positive ("North")). Surely that is easy to see.

Which quadrant R, the return route, is in is a little harder. Assuming your calculations are correct, since the x and y coordinate are both positive, it is in the first quadrant. However, it is hard to tell whether your R is correct because you don't say where you got the "168" and "261" that you use in your calculations. I suspect they are A+ B but, if so, that is incorrect. The location after the second leg is A+ B. The path necessary to fly back to the base camp is -(A+ B). That would be in the third quadrant.
 
  • #3
yes I've got them from A+B=R,
so R=-(A+B)->|R|=sqrt((-168^2)+(-261^2))=310km is the distance from lake B to base camp;
and if i calculate σ = arctan Ry/Rx->σ=57.2° south of west.
OK 30° west of north are 30° beetween y-axis and vector B^->, 30°+90°= 120°=2pi/3 in radiant.
 
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FAQ: Vector & Scalar Homework: Base Camp to Lake B

1. What is the difference between a vector and a scalar?

A vector has both magnitude and direction, while a scalar only has magnitude. For example, velocity is a vector because it includes both the speed and direction of an object, whereas temperature is a scalar because it only represents the magnitude of how hot or cold something is.

2. How do you calculate the magnitude of a vector?

The magnitude of a vector can be calculated using the Pythagorean theorem, where the magnitude is equal to the square root of the sum of the squared components. For example, if a vector has components of 3 and 4, its magnitude would be √(3² + 4²) = √25 = 5.

3. What is the difference between a displacement vector and a distance scalar?

A displacement vector is a directed line segment that represents the change in position from one point to another. It has both magnitude and direction. Distance scalar, on the other hand, only represents the total length of the path traveled without any regard for direction.

4. How do you add vectors?

To add two vectors, you first need to resolve them into their horizontal and vertical components. Then, you can add the corresponding components together to get the resultant vector. The magnitude of the resultant vector can be calculated using the Pythagorean theorem, and the direction can be found using trigonometric functions.

5. How is vector addition different from scalar addition?

In vector addition, both the magnitude and direction of the vectors need to be taken into account, while scalar addition only involves adding the numerical values without any regard for direction. Vector addition also follows the commutative and associative properties, while scalar addition follows only the commutative property.

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