Question

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

Short answer

The sketch of vectors in the vector field \( \mathbf{F}(x, y)=x \mathbf{i}+y \mathbf{j} \) gives insight into how a particle would move in this field. It's seen that the vectors point outward from the origin and their magnitude increases linearly as we move away from the origin. A particle would thus move in straight lines outward from the origin, accelerating as it moves further from origin.

Step-by-step solution

Understanding the Vector Field

The given vector field, \( \mathbf{F}(x, y)=x \mathbf{i}+y \mathbf{j} \), describes a vector's components along the x and y axis, respectively. A vector at a particular point (x, y) in the vector field can be represented by \( \mathbf{F}(x, y) \), which is simply the vector with x-components equal to x and y-components equal to y. Hence, it will be more of a linear field where the vectors increase in magnitude as we move away from the origin.

Begin Sketching the Vector Field

Start by sketching the vector at a few representative points. For example, let's pick the points (1, 0), (0, 1), (-1, 0), and (0, -1). At point (1, 0), the vector has its tail at the point (1,0) and points in the direction of the vector \( \mathbf{F}(1,0) = \mathbf{i} \), that is along the x-axis. Sketch this vector on the plane. Repeat this for the points (0, 1), (-1, 0), and (0, -1), sketching the respective vectors \( \mathbf{F}(0,1) = \mathbf{j} \), \( \mathbf{F}(-1,0) = -\mathbf{i} \), and \( \mathbf{F}(0,-1) = -\mathbf{j} \).

Continue Sketching More Vectors

Continue sketching vectors for more points in the plane. Try to include both positive and negative x and y coordinates, as well as points with x=y, to create a complete picture of the vector field. In the end, the pattern should be clear - the vectors point outward from the origin and their magnitude increase linearly as we move away from the origin in any direction. This visual representation gives a clear idea of how a particle would move in this vector field.