Q. 8 - Calculus

Question

Do the vectors in the range of $\mathbf{F}(x,y)=x\mathbf{i}+y\mathbf{j}$ point towards or away from the origin?

Step-by-Step Solution

Answer

The range of $\mathbf{F}(x,y)=x\mathbf{i}+y\mathbf{j}$ vector points away from the origin.

Step 1.Given Information

Do the vectors in the range of $\mathbf{F}(x,y)=x\mathbf{i}+y\mathbf{j}$ point towards or away from the origin?

Step 2. There are two types of vector fields in ℝ 2 : radial fields and rotational fields.

Radial fields model certain gravitational fields and energy source fields, and rotational fields model the movement of a fluid in a vortex. In a radial field, all vectors either point directly toward or directly away from the origin. Furthermore, the magnitude of any vector depends only on its distance from the origin. In a radial field, the vector located at point $(x,y)$ is perpendicular to the circle centered at the origin that contains point $(x,y)$, and all other vectors on this circle have the same magnitude.

Step 3. The range of F ( x , y ) = x i + y j vector points away from the origin

The vector field $\mathrm{F}(x,y)=(x,y)$ each vector points away from the origin, their magnitudes increase as you move away from the origin, and the magnitudes are constant on spheres centered at the origin.