Question

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

Short answer

The representative vectors at selected points in the vector field \(\mathbf{F}(x, y) = x \mathbf{i} - y \mathbf{j}\) can be sketched on the xy plane. The x-component of the vector at a point (x, y) is equal to x while the y-component is equal to -y. The vectors point in the respective directions and have magnitudes as per the x and y values at the points.

Step-by-step solution

Understand the given vector field

The given vector field is \(\mathbf{F}(x, y) = x \mathbf{i} - y \mathbf{j}\). Here, \(\mathbf{i}\) and \(\mathbf{j}\) are the unit vectors in the x and y direction respectively. For any general point (x, y), the vector field \(\mathbf{F}\) would denote a vector whose x-component is x and y-component is -y.

Choose representative points

Choose several representative points in the xy plane. We can select some integral points such as (0,0), (1,0), (0,1), (-1,0), (0,-1), (1,1), (-1,-1), (-1,1) and (1,-1).

Determine vectors at the representative points

Determine the vectors given by the vector field at these points. For example, at point (1,0), \(\mathbf{F}(1, 0) = 1 \mathbf{i} - 0 \mathbf{j} = \mathbf{i}\). Similar calculations for the rest of the points will provide the respective vectors.

Sketch the vectors

Sketch the vectors at their corresponding points. Draw them as arrows pointing in the direction of the vector and the length representing the magnitude of the vector. The collected vectors from Step 3 should help in making an accurate sketch.