Vector components/coordinates question.

[Physics345]

Homework Statement

The velocity of a plane is 625 km/h northwest.

a) Draw a diagram of this vector, including the magnitude and direction.
b) Determine the horizontal and vertical components of this vector. Round your answers to the nearest whole number.
c) Give the coordinates of the vector in Cartesian form.
d) Describe the direction of the vector in three other way.

Homework Equations

The Attempt at a Solution

For this question my issue is with question c) is it literally asking me to pretty much repeat what I found in b) but in a different way. It seems really, unnecessary when I view it in this manner, making me doubt the answer. Therefore making my answer for either b) or c) wrong or maybe I'm just over thinking it would be great to get some peace of mind if anyone has the time to confirm for me.
Anyways here's my work:

 

[Let'sthink]

I think using the term vertical and horizontal is not proper north component and west component should have been used.

 

[Physics345]

I think using the term vertical and horizontal is not proper north component and west component should have been used.

Um, I do not think that's true (I could be wrong, I'm no expert). My physics/calculus textbooks do the same thing. The horizontal refers, to the horizontal of the Cartesian plane that has been drawn above it, which can either be positive for down(south)/left(west) or up(north)/right(east) it is very clear where the horizontal/vertical is when it comes to 2 dimensional planes regardless of notation used. At least this is how I view it.

 

[SammyS]

I think using the term vertical and horizontal is not proper north component and west component should have been used.

I totally agree with @Let'sthink .

Admittedly, restating the problem in a "correct" form may make the wording awkward.

Asking you to use a Cartesian coordinate frame with positive x-axis aligned eastward and positive y-axis aligned northward would make things clear. The question could then ask for x and y components of the velocity vector.

 

[Physics345]

I totally agree with @Let'sthink .

Admittedly, restating the problem in a "correct" form may make the wording awkward.

Asking you to use a Cartesian coordinate frame with positive x-axis aligned eastward and positive y-axis aligned northward would make things clear. The question could then ask for x and y components of the velocity vector.

I did make the positive axis N & E though. Anyways isn't the answer correct the Pythagorean theorem confirms are 442. How exactly did u want me to re do it? and when I attempt the solution again, should I expect different answers? Also wouldn't it just be best If I rearranged the RA triangle to go WN instead, wouldn't that make my solution clear?

 

[SammyS]

I did make the positive axis N & E though. Anyways isn't the answer correct the Pythagorean theorem confirms are 442. How exactly did u want me to re do it? and when I attempt the solution again, should I expect different answers? Also wouldn't it just be best If I rearranged the RA triangle to go WN instead, wouldn't that make my solution clear?

No issue with your work. My issue was with the problem statement.

 

[Physics345]

Well I basically re did it in a slightly different manner, I feel like it is more clear this way. Take a look and let me know what you think:

 

[Let'sthink]

I have not checked numbers, but one thing I wish to state. if we right x =-442 then it is understood that it is 442 along west. Writing it x = -442[W] is superfluous.

 

[Physics345]

I have not checked numbers, but one thing I wish to state. if we right x =-442 then it is understood that it is 442 along west. Writing it x = -442[W] is superfluous.

True, that was a mistake I did not notice to be honest. Regardless your help was and is very much appreciated thank you very much.

 

[Physics345]

Other than that, which solutution would you say is better the first or second?

On a side note here's an updated version of the second solution:

 

[SammyS]

Other than that, which solutution would you say is better the first or second?

On a side note here's an updated version of the second solution:
View attachment 222579

Hey, they're both good.

The second uses angles referenced to the positive x-axis: good "math"-wise. However, it's a problem related to physics. I have known many students in physics classes who struggled with trigonometry based upon "opposite, adjacent, and hypotenuse" trigonometry.

 

[Physics345]

Hey, they're both good.

The second uses angles referenced to the positive x-axis: good "math"-wise. However, it's a problem related to physics. I have known many students in physics classes who struggled with trigonometry based upon "opposite, adjacent, and hypotenuse" trigonometry.

When I apply this type of math to physics it usually turns out slightly different, but over all the same depending on what the question is asking. I feel as though physics, is more of understanding what's really going on in the question at hand plus visualization. Anyways I'm glad I don't have that issue I normally look at all sections of the question at hand to make sure I'm drawing a proper FBD, which is more important than anything since there could be more than a few triangles and multiple forces acting on a single object. The biggest Issue I feel with other people is not fully understanding the basics of algebra and trig before advancing into more advanced math. Anyways I feel like this topic has been discussed to it's full length and I have succeeded in my goal of understanding this section. At this point I just need to practice more which I always do since math is fun! Once again I appreciate the help of both you and Letsthink, Thank you very much for taking your time to help me out.

 

[Let'sthink]

Anyways I feel like this topic has been discussed to it's full length and I have succeeded in my goal of understanding this section. At this point I just need to practice more which I always do since math is fun! Once again I appreciate the help of both you and Letsthink, Thank you very much for taking your time to help me out.

First, but for some notation issues I agree with Sammy that both methods are fairly good. But I do not very much agree to call one method as physics and other maths. Mathematics is the technique which is used in physics also it is said mathematics is language of physics. Just as a "thought" expressed in a language is both a thought and also language item. I would at best say they are two different uses of mathematics algebra, trigonometry and use of a pythagorean triangle to show a vector to be the sum of two chosen vectors in mutually perpendicular directions.

Restricting myself to only in a plane, for every vector (having an initial point and the tip) it can be considered as a sum of two vectors by constructing a parallelogram or by constructing a triangle. These refer to parallelogram law of vector addition and triangle law of vector addition. Parallelogram law can be directly verified by use of weights and pulleys and a thought experiment of taking a particle through successive displacements intuitively verifies the triangle law of addition. Even by using a diagram one can prove that parallelogram law can be derived assuming triangle law to be true. But for a given vector we will have infinite number of such pairs which will add up to the given vector. To limit this to, so called resolution of a vector into two fixed vectors we fix some direction and think about the possible vector pair one along the chosen direction and another perpendicular to it. First direction is chosen in such a way that the given vector makes an acute angle with the given vector(direction). Simple trigonometry tells us that one component will have magnitude cos theta times the magnitude of the given vector and the other will be sin theta times the magnitude.

But in mathematics the trigonometric functions are defined not only for acute angles but all angles ranging up to infinity in a periodic way of course. That technique then can be used to generalize cos theta component and sine theta component for all theta and one does not have to draw a triangle for that. Your first method requires drawing a triangle and second method does not require drawing of a triangle. Only you must decide what is positive x-axis and the sense of anticlockwise direction as positive theta then everything else including the y-axis is fixed. [Although drawing of the triangle is included in defining the trigonometric functions for all angles and you do draw triangle in different quadrants. But as user of mathematics technique we do not have to go into that.]

Finally coming to the notation issue. you write 442 = (vector y)/|r|; I think by vector y you mean y component of vector. But notation-wise writing that way is not proper as on left side we have scalar and on right we see vector. a scalar cannot be equal to a vector.