Reading More Than One Arrow
A vector field assigns a vector to each point in its domain. A field diagram shows selected values of that assignment as arrows: the arrow’s tail marks the point where the vector is assigned, its orientation shows the vector’s direction, and its length represents its magnitude according to the diagram’s scale.
Comparing arrows at different points means asking two separate questions. Are their magnitudes different? Are their directions different? An arrow can become longer without turning, turn without changing length, or do both. Keeping these observations separate makes a field diagram easier to read accurately.
For a two-dimensional field \(\mathbf{F}\), the vector assigned at \((x,y)\) is written \(\mathbf{F}(x,y)\). If \(\mathbf{F}(x,y)=\langle P(x,y),Q(x,y)\rangle\), then its magnitude is \(\sqrt{P(x,y)^2+Q(x,y)^2}\). The direction depends on the relative signs and sizes of the components, not just on the magnitude.
Length and Orientation Must Be Read Separately
When arrows are drawn to a common scale, a longer arrow represents a vector with greater magnitude. If one arrow is twice as long as another, its vector magnitude is twice as large. But the arrow’s orientation is a separate feature: arrows of different lengths may point in the same direction, and arrows of equal lengths may point in different directions.
A diagram may include a scale such as “one centimetre represents two units of magnitude.” In that case, the measured length on the page is not itself the vector magnitude. Multiply the drawn length by the scale factor. If the arrows are not drawn to a common scale, their visual lengths cannot be compared reliably unless the diagram gives enough information to correct for the different scales.
Worked Example: Comparing Vectors at Four Points
Worked Example: Comparing Vectors at Four Points
Suppose a sampled field has these values:
| Point | Assigned vector |
|---|---|
| \(A\) | \(\mathbf{F}(A)=\langle 3,4\rangle\) |
| \(B\) | \(\mathbf{F}(B)=\langle 6,8\rangle\) |
| \(C\) | \(\mathbf{F}(C)=\langle -4,3\rangle\) |
| \(D\) | \(\mathbf{F}(D)=\langle 0,0\rangle\) |
First calculate each magnitude using the component formula.
Thus the vector at \(B\) has twice the magnitude of the vector at \(A\). The vectors at \(A\) and \(C\) have equal magnitudes, and the vector at \(D\) has zero magnitude.
To compare directions, find the unit direction vectors for the nonzero vectors:
So the vectors at \(A\) and \(B\) point in the same direction. For \(C\), the unit direction vector is \(\langle -4/5,3/5\rangle\), which differs from both. In fact, the dot product of the vectors at \(A\) and \(C\) is \(3(-4)+4(3)=-12+12=0\), so these directions are perpendicular. At \(D\), there is no direction to compare because the vector is zero.
Check: Multiplying each nonzero unit direction by its magnitude recovers its original vector. For example, \(5\langle 3/5,4/5\rangle=\langle 3,4\rangle\), and \(10\langle 3/5,4/5\rangle=\langle 6,8\rangle\). The magnitude and direction comparisons are consistent with all four assigned vectors.
What It Means for a Field to Change
When describing a diagram, say what happens to magnitude and direction separately. For example, moving across a row of sampled points, you might observe that the arrows grow longer while continuing to point east. That describes increasing magnitude and unchanged direction across those samples. If the arrows remain the same length but rotate from east toward north, the observed direction changes while the magnitude remains the same.
A field diagram usually displays only selected points. It supports conclusions about those arrows and, when the pattern is clear, a cautious description of the displayed region. A few samples alone do not prove what happens at every point in the domain. Use wording such as “the sampled arrows grow longer from left to right” rather than claiming a global rule that the diagram does not establish.
Worked Example: Converting Drawn Lengths with a Scale
Worked Example: Converting Drawn Lengths with a Scale
In a field diagram, suppose the key says that \(1\) centimetre of drawn arrow length represents \(2\) magnitude units. Three arrows have measured lengths of \(2\) centimetres, \(3.5\) centimetres, and \(1\) centimetre. The first and second point \(30^\circ\) above the positive horizontal direction; the third points directly left.
Convert each drawn length into a vector magnitude by multiplying by \(2\) magnitude units per centimetre:
The second vector has the greatest magnitude, \(7\) units; the first has magnitude \(4\) units; and the third has magnitude \(2\) units. The first two arrows have the same direction, although the second is longer. The third points in the opposite horizontal direction from the positive horizontal axis, so it does not share their orientation.
Check: Dividing the calculated magnitudes by the scale factor gives the original drawn lengths: \(4/2=2\) centimetres, \(7/2=3.5\) centimetres, and \(2/2=1\) centimetre. This confirms that the scale was applied in the correct direction.
Why the Magnitude Formula Works
The formula for magnitude follows from the geometry of the components. For \(\mathbf{v}=\langle a,b\rangle\), place its tail at the origin. Its tip is at \((a,b)\). The horizontal and vertical displacements form the legs of a right triangle, with lengths \(|a|\) and \(|b|\); the vector is the hypotenuse. By the Pythagorean theorem, the square of the hypotenuse is \(a^2+b^2\). Since magnitude is nonnegative, taking the nonnegative square root gives the formula.
Dividing a nonzero vector by its magnitude keeps its orientation and gives it length one. Indeed, if \(\mathbf{u}=\mathbf{v}/\|\mathbf{v}\|\), then
This is why unit direction vectors let us compare orientations without being distracted by differences in arrow length. The construction requires \(\mathbf{v}\ne\mathbf{0}\), because division by its magnitude is undefined when the magnitude is zero.
Worked Example: Describing Change in a Simple Field
Worked Example: Describing Change in a Simple Field
Consider the field \(\mathbf{F}(x,y)=\langle 2,y\rangle\), sampled at the points \((0,0)\), \((0,1)\), and \((0,2)\). Substitute each point into the formula:
Their magnitudes are
The arrows get longer across these samples because \(2<\sqrt{5}<2\sqrt{2}\). Their directions also change: the first points horizontally right, while the next two have positive vertical components and point above the horizontal. The unit direction vectors confirm this:
Check: Multiplying each unit direction vector by its corresponding magnitude gives \(\langle 2,0\rangle\), \(\langle 2,1\rangle\), and \(\langle 2,2\rangle\), respectively. The calculations agree with the field values. These three samples show increasing magnitude and a changing orientation along the sampled points; by themselves, they are not a proof about every possible path through the field.
A Common Mistake: Adding Components to Get Length
A frequent error is to treat the magnitude of \(\langle 3,4\rangle\) as \(3+4=7\). That sum is not the arrow’s length. The components are perpendicular displacements, so their relationship to the length is given by the Pythagorean theorem, not ordinary addition.
Check: A vector of magnitude \(5\) with components \(3\) and \(4\) satisfies \(3^2+4^2=5^2\), as required by the right-triangle relationship. The incorrect value \(7\) does not satisfy that relationship.
Describe What the Diagram Shows
A clear comparison names the points or region being compared, reports magnitude and direction separately, and respects the diagram’s scale. Avoid inferring a continuous trend from a small number of arrows unless the field’s formula or a sufficiently informative diagram supports that conclusion. These habits make descriptions precise whether the arrows are given as component vectors, drawn to scale, or shown as samples on a grid.
Check Your Understanding
Use the definitions and examples above to answer the following questions.
- For \(\mathbf{v}=\langle 5,12\rangle\), find its magnitude and unit direction vector.
- Two nonzero arrows have equal lengths but different orientations. What can you conclude about their magnitudes and directions?
- A diagram uses the scale \(1\) centimetre for \(3\) magnitude units. What magnitude does a \(2.5\)-centimetre arrow represent?
- Why does the zero vector have no unit direction vector?
- What can three sampled arrows establish about a trend, and what can they not establish by themselves?