Question

Sketch several representative vectors in the vector field. $$ \mathbf{F}(x, y, z)=3 y \mathbf{j} $$

Short answer

The solution to this exercise is not numerical but graphical. It requires drawing several representative vectors of the vector field \( \mathbf{F}(x, y, z)=3y \mathbf{j} \) which are only along the y-axis, their magnitude being thrice their y-coordinate.

Step-by-step solution

Understanding the Vector Field

In a three dimensional space, a vector field associates to each point of this space, a vector. The given vector field \( \mathbf{F}(x, y, z)=3y \mathbf{j} \) instructs that the vector at any point P(x, y, z) has no component on the x-axis and the z-axis, i.e., it's exclusively in the y-direction. It lies entirely in the y-axis plane, with its magnitude being 3 times the y-coordinate.

Sketching the Vectors

The vector field is two-dimensional because the x and z components of the vectors remain consistent (at zero). It can be represented in a plane (x-y, y-z, or x-z). It is simplest to take it in the y-z plane because the vector field in question is exclusively directed along the y-axis. When y=0, the vector field will show vectors with length 0, when y=1, the vectors would be three times in length, and when y=-1, the vectors will be three times in length but in the opposite direction. Draw several representative vectors along the y-axis, in both the positive and negative directions, keeping their lengths proportional to their y-coordinates.

Double-check Your Sketch

Make sure that all the vectors are along the y-axis, and their lengths are proportional to their y-coordinates. Remember, vectors in the negative y direction should be represented with appropriate negativity (usually downwards or to the left depending on the orientation of your graph).