What Does a Unit Vector Field Look Like in Cartesian Coordinates?

In summary, the speaker is seeking help with a problem on vector calculus. They have solved parts (a) and (b) by plugging in a point and using the formula for S and the magnitude of S respectively. They are now asking for assistance with part (c) and are prompted to consider what the locus of points would look like mathematically when the magnitude of S is equal to 1.
  • #1
bibo_dvd
37
0
Hello guys !

i have this problems while solving problems on vector calculus ..

Fps0q98.png



i solved (a) , (b)
as i put P(2,4,3) in the formula of S and i solved it and i solved (b) as a(S)= S/lSl

but in (c) i don't how what should i do to solve it ..please help me guys ..Thank you
 
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  • #2
As you have solved b you must have already computed |S|. From having that expression you should be able to progress.
 
  • #3
Where the magnitude of S is 1, what does that look like mathematically in terms of x y and z? In other words, you have a locus of points, and all of these points, when you calculate the magnitude of S there, it's equal to 1. What does this look like mathematically?
 

FAQ: What Does a Unit Vector Field Look Like in Cartesian Coordinates?

What is a vector field?

A vector field is a mathematical concept that assigns a vector, such as velocity or force, to each point in a given space. It is often represented visually using arrows to show the direction and magnitude of the vector at each point.

What are some common problems that can occur with vector fields?

Some common problems with vector fields include divergence, curl, and singularities. Divergence occurs when the vectors at each point in the field are spreading out or converging, rather than remaining constant. Curl is when the vectors in the field start to rotate around a specific point. Singularities are points where the vector field becomes undefined or infinite.

How can I determine if a vector field is conservative?

A vector field is considered conservative if the work done by the field when moving from one point to another is independent of the path taken. This means that the total work done around a closed loop is equal to zero. To determine if a vector field is conservative, you can use the curl or divergence tests.

What are some real-life applications of vector fields?

Vector fields have many practical applications in various fields of science and engineering. Some common examples include fluid dynamics, where vector fields are used to represent the flow of liquids or gases; electromagnetism, where vector fields are used to represent the electric and magnetic fields; and economics, where vector fields are used to model supply and demand in markets.

How can I visualize a vector field?

There are various ways to visualize a vector field, including using arrows, color-coding, and streamlines. Arrows are commonly used to represent the direction and magnitude of the vectors at each point. Color-coding can also be used to show the strength of the vectors at each point. Streamlines are curved lines that follow the direction of the vectors, giving a sense of the flow of the field.

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