Question

Short answer

Step-by-Step Solution

Step-by-step solution

Concept

To sketch a vector field, draw the arrow representing the vector\({\rm{F}}\left( {{\rm{x,y,z}}} \right){\rm{ = xk}}\)starting at point\(\left( {{\rm{x, y, z}}} \right)\).

It is impossible to do this for all points\(\left( {{\rm{x, y, z}}} \right)\)but we can gain a reasonable impression of\({\rm{F}}\)by doing it for a few representative points in the domain.

Given Information.

The given vector field is \(F\left( {x,y,z} \right) = xk\).

Calculation.

The vector field on \({\mathbb{R}^3}\) is a function \(F\), that assigns to each point \(\left( {x,y,z} \right)\) in \(E\), which is a three dimensional vector \(F\left( {x,y,z} \right)\).

All vectors are vertical and length of vector field \(F\left( {x,y,z} \right)\) is \(\left| x \right|\).

For \(x > 0\), all points are moving upward direction and for \(x < 0\), all points are moving downward direction.

Sketch the vector field.

The sketch of the vector field will be: