From Component Functions to Vector Arrows

From a Formula to an Arrow

A vector field does more than assign a vector to each point: its component functions tell us exactly which vector belongs at any chosen point. To draw that value as an arrow, we need to keep two pieces of information separate. The point specifies where the arrow begins; the field’s components specify the arrow’s displacement from that beginning.

This distinction is easy to lose when writing coordinates. If a field assigns the vector \(\langle 2,-1\rangle\) at the point \((3,4)\), the arrow begins at \((3,4)\), not at \((2,-1)\). Its components describe how far to move horizontally and vertically from its starting point. We can use those movements to find the arrow’s endpoint, direction, and length.

Evaluating the Component Functions

A two-dimensional vector field can be written in component form as \(\mathbf{F}(x,y)=\langle P(x,y),Q(x,y)\rangle\), where \(P\) and \(Q\) are scalar-valued component functions. At a point \((a,b)\), evaluate both functions using \(x=a\) and \(y=b\). The result, \(\mathbf{F}(a,b)=\langle P(a,b),Q(a,b)\rangle\), is the vector assigned to that point.

In three dimensions, the same process applies to three component functions: \(\mathbf{F}(x,y,z)=\langle P(x,y,z),Q(x,y,z),R(x,y,z)\rangle\). At \((a,b,c)\), the assigned vector is \(\mathbf{F}(a,b,c)=\langle P(a,b,c),Q(a,b,c),R(a,b,c)\rangle\). Each component gives the displacement in one coordinate direction.

Evaluating a field means substituting the point’s coordinates into each component function—not treating the point’s coordinates as the vector itself. Once the vector is evaluated, it can be drawn as an arrow based at that point.

Turning Components into a Placed Arrow

For a point \((a,b)\) and an assigned vector \(\langle u,v\rangle\), the arrow begins at \((a,b)\) and ends at \((a+u,b+v)\). The vector’s first component \(u\) gives the horizontal change; its second component \(v\) gives the vertical change. Positive components move in the positive coordinate direction, and negative components move in the negative coordinate direction.

The endpoint is useful for drawing, but it is not the same thing as the assigned vector. The endpoint records the final location of the arrow’s head. The vector records the change from tail to head. To recover the vector from the drawn arrow, subtract the tail coordinates from the head coordinates.

The direction is determined by the components together. For instance, \(\langle 2,-5\rangle\) points right and down: the positive first component moves right, and the negative second component moves down. It does not point in the direction of either component alone.

In a coordinate diagram with equal scales on both axes, the length of a two-dimensional arrow with components \(\langle u,v\rangle\) is \(\sqrt{u^2+v^2}\). This follows from the Pythagorean theorem: the horizontal and vertical displacements form the perpendicular legs of a right triangle, while the arrow is its hypotenuse. In three dimensions, the corresponding length is \(\sqrt{u^2+v^2+w^2}\).

A General Rule and Its Justification

The placement rule follows directly from what vector components mean: they measure coordinate changes, not a new location. The same reasoning applies in any dimension. Write the starting point as \(\mathbf{p}=\langle a,b,c\rangle\) and the assigned vector as \(\mathbf{v}=\langle u,v,w\rangle\). Adding the component changes gives the head position \(\mathbf{p}+\mathbf{v}=\langle a+u,b+v,c+w\rangle\). Subtracting the tail from this head recovers the assigned vector.

Proof. Let the coordinates of \(\mathbf{p}\) be \(p_1,\ldots,p_n\), and let the components of \(\mathbf{F}(\mathbf{p})\) be \(v_1,\ldots,v_n\). By definition of component displacement, the arrow’s head has coordinates \(p_1+v_1,\ldots,p_n+v_n\). Subtracting the tail coordinates gives \((p_1+v_1-p_1,\ldots,p_n+v_n-p_n)=(v_1,\ldots,v_n)=\mathbf{F}(\mathbf{p})\). Therefore the arrow represents the assigned vector and is anchored at the specified point.

Direction, Length, and Scale

Components give a precise description of direction without requiring a sketch. In two dimensions, the signs tell which way the arrow moves along each axis, while the component magnitudes tell how much it moves along each axis. If one component is zero, there is no displacement in that coordinate direction. If both components are zero, the assigned vector has length zero and has no direction determined by its components.

When drawing, the length formula gives the vector’s mathematical length in coordinate units. A diagram may use a scale to represent that length—for example, one centimetre per two coordinate units—so the drawn segment need not have the same numerical length as the vector. Even with a scale, the tail must be placed at the point where the field was evaluated, and the arrow’s direction and relative length should reflect the assigned vector.

Common Mistake: Confusing the Vector with Its Location

A frequent error is to draw the evaluated vector from the origin even when the field was evaluated elsewhere. For the first example, the field value at \((1,2)\) is \(\langle 0,5\rangle\). Drawing an arrow from the origin to \((0,5)\) uses that vector as a free arrow, but it does not show the field arrow at the requested point.

The correction is to anchor the arrow at \((1,2)\). Its head is \((1,2)+\langle 0,5\rangle=(1,7)\), so the field arrow runs from \((1,2)\) to \((1,7)\). Subtracting the tail from the head gives \((1-1,7-2)=(0,5)\), confirming that it represents the same vector while being placed at the correct point.

Another useful check is to avoid using the head coordinates as though they were the vector components. In that same example, the head is \((1,7)\), but the assigned vector is \(\langle 0,5\rangle\). The head tells where the arrow ends; subtracting its starting point recovers the vector.

Why This Representation Matters

Calculating a field value and placing its arrow are two connected but distinct tasks. The component functions determine the vector at a point; the point determines where its arrow is anchored. Keeping those roles separate makes it possible to interpret formulas accurately and check a diagram against the calculation.

This tutorial concerns one evaluated point at a time. For any such point, the procedure is exact: evaluate every component, place the tail at the point, add the vector components to find the head, and verify the result by subtraction. The length follows from the Euclidean norm, while the signs and sizes of the components describe the arrow’s coordinate-wise motion.