Sketch the vector field F⃗ (r⃗ )=2r⃗ in the plane, where r⃗ =⟨x,y⟩. Select all that apply.
A.All the vectors point away from the origin.
B. The vectors increase in length as you move away from the origin.
C. All the vectors point toward the origin.
D. The vectors decrease in length as you move away from the origin.
E. All the vectors point in the same direction.
F. The length of each vector is 2.
Looks like A&B to me.
I mean, have you tried actually drawing a vector xi+yj, with its tail at (x,y)?
A. All the vectors point away from the origin.
E. All the vectors point in the same direction.
F. The length of each vector is 2.
The vector field F⃗ (r⃗ )=2r⃗ , where r⃗ =⟨x,y⟩, can be visualized by plotting vectors at different points in the plane. To determine the properties of the vector field, let's analyze each option:
A. All the vectors point away from the origin.
This is not true. The vectors point radially outward from the origin, but they also point inward as you move away from the origin in the opposite direction.
B. The vectors increase in length as you move away from the origin.
This is true. The length of the vectors increases as you move away from the origin.
C. All the vectors point toward the origin.
This is not true. As mentioned earlier, the vectors point radially outward and inward.
D. The vectors decrease in length as you move away from the origin.
This is not true. The length of the vectors increases as you move away from the origin.
E. All the vectors point in the same direction.
This is not true. The vectors point radially outward and inward from the origin.
F. The length of each vector is 2.
This is not true. The length of each vector depends on the coordinates of the point but is not constant.
So, the correct options are B. The vectors increase in length as you move away from the origin.
To sketch the vector field F⃗ (r⃗ )=2r⃗ in the plane, we need to determine the direction and magnitude of the vectors at different points (x, y).
The vector field F⃗ (r⃗ )=2r⃗ can be written as F⃗ (x, y) = 2⟨x, y⟩.
Let's consider a few points to determine the direction and magnitude of the vectors:
1. At the origin (0, 0), F⃗ (0, 0) = 2⟨0, 0⟩ = ⟨0, 0⟩. This means the vector at the origin is the zero vector, which has no direction or magnitude.
2. For points in the positive x-axis (x > 0), F⃗ (x, 0) = 2⟨x, 0⟩ = ⟨2x, 0⟩. This means the vectors are pointing in the positive x-direction and have a magnitude of 2x. As x increases, the vectors increase in length.
3. For points in the negative x-axis (x < 0), F⃗ (x, 0) = 2⟨x, 0⟩ = ⟨2x, 0⟩. This means the vectors are pointing in the negative x-direction and have a magnitude of 2x. However, as x decreases (moves away from the origin), the vectors decrease in length.
4. For points in the positive y-axis (y > 0), F⃗ (0, y) = 2⟨0, y⟩ = ⟨0, 2y⟩. This means the vectors are pointing in the positive y-direction and have a magnitude of 2y. As y increases, the vectors increase in length.
5. For points in the negative y-axis (y < 0), F⃗ (0, y) = 2⟨0, y⟩ = ⟨0, 2y⟩. This means the vectors are pointing in the negative y-direction and have a magnitude of 2y. However, as y decreases (moves away from the origin), the vectors decrease in length.
6. For points not on the coordinate axes (x ≠ 0, y ≠ 0), F⃗ (x, y) = 2⟨x, y⟩. These vectors will point in the direction of the coordinate (x, y) and have a magnitude of 2 times the length of the vector ⟨x, y⟩.
Based on these observations, we can conclude the following:
A. Not all the vectors point away from the origin because vectors on the negative x-axis and negative y-axis point toward the origin.
B. The vectors increase in length as you move away from the origin, except for the vectors on the negative x-axis and negative y-axis which decrease in length.
C. The vectors on the positive x-axis and positive y-axis point away from the origin, but the vectors on the negative x-axis and negative y-axis point toward the origin.
D. The vectors decrease in length as you move away from the origin on the negative x-axis and negative y-axis, but increase in length for other points.
E. The vectors do not point in the same direction as they vary in direction based on the location.
F. The length of each vector is not constant and varies with the coordinates of each point.
Therefore, the only correct statement is B. The vectors increase in length as you move away from the origin.