Vector Components as Directed Quantities

Introduction

A point in three-dimensional space tells us where something is. A vector describes a directed change: how far an arrow goes along each coordinate axis, with each change recorded as a signed number.

This distinction matters because an arrow can be drawn in different places while representing the same directed quantity. The vector is not the location of its tail or head. It is the three-component description of the arrow’s displacement.

For now, the emphasis is on what the components mean and what they do not mean. Later work will develop systematic ways to read signs, find a vector from two endpoints, and calculate vector magnitudes in three dimensions.

Points and Vectors Describe Different Things

A point \(P=(x,y,z)\) identifies a location using its coordinates relative to the coordinate axes. A vector \(\mathbf v=\langle a,b,c\rangle\), by contrast, records three signed coordinate changes. In order, these are the changes along the \(x\)-, \(y\)-, and \(z\)-axes.

Definition: A three-dimensional vector is an ordered triple \(\mathbf v=\langle a,b,c\rangle\) of signed components. When represented by an arrow, its components describe the change from the arrow’s tail to its head along the three coordinate axes. The point coordinates of the tail and head depend on where the arrow is placed; the components describe the directed quantity represented by the arrow.

The components are signed: they can be positive, negative, or zero. A zero component means there is no change along its corresponding axis. The ordered triple matters because each position has a fixed coordinate-axis role: first for \(x\), second for \(y\), third for \(z\).

An arrow is a useful picture of a vector. Its tail marks where the arrow begins, and its head marks where it ends. The vector’s components describe the directed change from tail to head. Moving the whole arrow to another location changes the coordinates of its tail and head, but it need not change the changes that the arrow represents.

A Vector Is Not an Endpoint

Suppose a point has coordinates \((1,4,-2)\). Those three numbers locate the point; they do not, by themselves, describe an arrow. If an arrow with vector \(\langle 3,-2,1\rangle\) begins at that point, its head is at \((4,2,-1)\). The point \((4,2,-1)\) is the arrow’s endpoint, while \(\langle 3,-2,1\rangle\) is the directed change it represents.

The same distinction holds even when a vector is drawn with its tail at the origin. In that special drawing, the head’s coordinates happen to match the vector’s components. The match is a convenient way to picture the vector, not a definition that makes every vector an endpoint.

Worked Example: One Vector, a Tail Away from the Origin

Consider the vector \(\mathbf v=\langle 3,-2,1\rangle\) represented by an arrow whose tail is \(P=(1,4,-2)\). Find the head \(Q\), and distinguish its coordinates from the vector’s components.

The vector gives the changes from tail to head: \(3\) in the first coordinate, \(-2\) in the second, and \(1\) in the third. Add these changes to the tail coordinates, one coordinate at a time:

Q=(1+3,\;4+(-2),\;-2+1)=(4,2,-1).

Thus the head is the point \(Q=(4,2,-1)\). Its coordinates are not the components of \(\mathbf v\). Check the changes from the stated tail to the computed head:

4-1=3,\qquad 2-4=-2,\qquad -1-(-2)=1.

These are exactly the three components of \(\mathbf v=\langle 3,-2,1\rangle\). The check confirms that \(Q\) is the correct endpoint for this arrow, while the vector remains the directed change.

Why Moving an Arrow Does Not Change Its Vector

The components record changes, not absolute positions. Suppose an arrow begins at \(P=(x_1,y_1,z_1)\) and ends at \(Q=(x_2,y_2,z_2)\). Its component triple is the change in each coordinate from \(P\) to \(Q\). If both points are shifted by the same amount in each coordinate, their coordinate changes stay the same.

Key result: Translating an arrow—moving its tail and head by the same displacement—does not change the vector represented by that arrow. The tail and head locations change, but the three coordinate changes from tail to head remain fixed.

Here is a direct justification. Let the same shift of both endpoints be described by the coordinate amounts \(r\), \(s\), and \(t\). Before the shift, the first component is \(x_2-x_1\). After the shift, the first component is \((x_2+r)-(x_1+r)\), which simplifies to \(x_2-x_1\). The common shift cancels. The same cancellation occurs in the second and third coordinates.

\begin{aligned} (x_2+r)-(x_1+r)&=x_2-x_1,\\ (y_2+s)-(y_1+s)&=y_2-y_1,\\ (z_2+t)-(z_1+t)&=z_2-z_1. \end{aligned}

Therefore the translated arrow has the same ordered triple of components as the original arrow. This is why a vector can be represented by arrows in different locations: location is not one of the vector’s components.

Worked Example: Shifting an Arrow Without Changing Its Components

An arrow begins at \(A=(0,1,2)\) and ends at \(B=(2,4,1)\). Both endpoints are then shifted by \((3,-1,2)\). Find the shifted endpoints and verify that the arrow’s components remain unchanged.

Adding the same shift to each endpoint gives

A'=(0+3,\;1+(-1),\;2+2)=(3,0,4),

B'=(2+3,\;4+(-1),\;1+2)=(5,3,3).

For the original arrow, the changes from \(A\) to \(B\) are

(2-0,\;4-1,\;1-2)=(2,3,-1).

For the shifted arrow, the changes from \(A'\) to \(B'\) are

(5-3,\;3-0,\;3-4)=(2,3,-1).

The components match. As a check on the shift itself, the first endpoint moved by \((3,-1,2)\), and the second endpoint also moved by \((3,-1,2)\). Because both moved equally, their relative directed change did not change. The vector is \(\langle 2,3,-1\rangle\) in either drawing.

Direction and Length Belong to the Vector

A nonzero vector has a direction and a length, also called its magnitude. Its direction is the way the arrow points, and its magnitude is the length of the arrow. These features describe the vector, not the coordinates of a point where the arrow happens to end.

The location of a drawn arrow can change without changing its direction or length, provided the arrow is shifted without being turned or stretched. Conversely, the coordinates of an endpoint alone do not tell you the length or direction of an arrow unless the tail location is also known.

Important qualification: The zero vector \(\mathbf 0=\langle 0,0,0\rangle\) has magnitude \(0\) and no well-defined direction. Since it has no nonzero displacement, there is no direction in which it points. The claim that a vector has both direction and length applies to nonzero vectors.

Worked Example: Length and Direction Are Not Endpoint Coordinates

Draw the vector \(\mathbf w=\langle 3,4,0\rangle\) with its tail at the origin. Identify its head and find its length using the Pythagorean theorem.

With the tail at \((0,0,0)\), the components give a head at \((3,4,0)\). Because the third component is zero, the arrow lies in the \(xy\)-plane. Its length is the hypotenuse of a right triangle with legs of lengths \(3\) and \(4\):

\|\mathbf w\|=\sqrt{3^2+4^2} =\sqrt{9+16} =\sqrt{25} =5.

The vector’s direction is the direction of the arrow from the origin toward \((3,4,0)\); its length is \(5\). The triple \((3,4,0)\) is the head’s location in this particular drawing, while \(\langle 3,4,0\rangle\) describes the directed change. To check the length calculation, the right-triangle relation gives \(3^2+4^2=9+16=25=5^2\).

If the same arrow were shifted so its tail were not at the origin, the head’s coordinates would change, but the vector’s directed change and length would still be \(\langle 3,4,0\rangle\) and \(5\).

The Zero Vector: A Special Case

The vector with all three components zero represents no change in any coordinate. If drawn as an arrow, its tail and head coincide. Its magnitude is zero, but it does not point along any direction. This is different from a short arrow: even a very short nonzero arrow has a direction, while the zero vector does not.

Worked Example: A Vector with No Direction

Let \(\mathbf u=\langle 0,0,0\rangle\), drawn with its tail at \(R=(2,-1,5)\). Determine its head and explain what can be said about its direction and magnitude.

There is no change in any coordinate, so the head is still \(R\):

(2+0,\;-1+0,\;5+0)=(2,-1,5).

The arrow has length \(0\), since its tail and head are the same point. Its direction is not defined: no nonzero arrow is present to point in a particular direction. As a check, each coordinate change is zero: \(2-2=0\), \(-1-(-1)=0\), and \(5-5=0\), matching \(\mathbf u\).

A Common Mistake: Treating the Head as the Vector

A frequent error is to identify the head’s coordinates with the vector, even when the arrow does not begin at the origin. For instance, in the first example the head was \((4,2,-1)\). It would be wrong to say that the vector was \(\langle 4,2,-1\rangle\) just because those are the endpoint coordinates.

The correction is to keep the roles separate. The tail was \((1,4,-2)\), the head was \((4,2,-1)\), and the vector recorded the changes between them:

(4-1,\;2-4,\;-1-(-2))=(3,-2,1).

Thus the vector is \(\langle 3,-2,1\rangle\), not the head’s coordinates. The mistaken triple would describe a different directed change, not the arrow shown. Endpoint coordinates identify locations; vector components identify signed changes.

Why This Viewpoint Matters

Thinking of components as directed quantities makes it possible to separate an arrow’s essential information from the place where it is drawn. This is useful in geometry, physics, and later vector calculations: a displacement can be represented wherever it is convenient, while its components continue to describe the same change.

When interpreting a diagram or a formula, ask what each triple represents. Is it a point locating a tail or a head? Or is it a vector describing an arrow’s three coordinate changes? Keeping that question in view prevents the central confusion of this topic.

Identify the object.
A point gives a location; a vector gives a directed change.

Keep the component order.
The three entries correspond, in order, to changes along the \(x\)-, \(y\)-, and \(z\)-axes.

Separate location from vector properties.
The endpoint’s coordinates depend on where the arrow is placed; a nonzero vector’s direction and magnitude describe the arrow itself.

Check Your Understanding

Answer these questions using the distinction between locations and directed changes.

What does each component in an ordered triple for a three-dimensional vector represent?
Why can translating an arrow change its endpoint coordinates without changing its vector?
An arrow begins at \((2,0,1)\) and has components \(\langle 1,-3,2\rangle\). What is its head, and why is that point not the vector?
What are the magnitude and direction status of the zero vector?
Why do the endpoint coordinates of an arrow drawn from the origin happen to match its components?