Position Vectors and Displacements

Distinguish a point from its position vector and a position vector from a movable displacement. Use starts, endpoints, origins, axes, and units to interpret the same coordinates correctly.

A point names a location, while a displacement vector describes a change from one location to another. Coordinates can represent either idea, but the interpretation must identify an origin or starting point before the numbers have a geometric meaning.

A Position Vector Starts at the Origin

Choose an origin OO and two coordinate axes. If a point PP is three units along the first axis and two units along the second, its coordinates are:

P=(3,2).P=(3,2).

The position vector of PP is the vector from the origin to that point:

OP→=[3,2].\overrightarrow{OP}=[3,2].

The point PP is a location. The vector OP→\overrightarrow{OP} is the directed change that reaches that location from the chosen origin. In elementary coordinate calculations, both are often written with the same pair of numbers, but they answer different questions.

Exercise: Read a position vector

Relative to an origin OO, a point is located at P=(−2,5)P=(-2,5). What is its position vector OP→\overrightarrow{OP}?

Compute it first, then check your number.

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HintBegin at the origin

The position vector records the horizontal and vertical changes needed to reach PP from OO.

SolutionUse the point coordinates

Starting at O=(0,0)O=(0,0), reaching P=(−2,5)P=(-2,5) requires a change of −2-2 along the first axis and 55 along the second. Therefore:

OP→=[−2,5].\overrightarrow{OP}=[-2,5].

A Displacement Connects Two Points

Suppose an object starts at A=[1,2]A=[1,2] and moves by the displacement:

d=[3,−1].\mathbf{d}=[3,-1].

The first coordinate moves three units in the positive first-axis direction. The second moves one unit in the negative second-axis direction. A displacement records both the direction and the amount of change. Calling it only a direction can hide the fact that [3,−1][3,-1] and [6,−2][6,-2] point the same way but describe different amounts of movement.

Combining the start and the displacement gives the endpoint:

B=A+d=[1,2]+[3,−1]=[4,1].B=A+\mathbf{d} =[1,2]+[3,-1] =[4,1].

Here, the equation records the relationship among a starting point, a displacement, and an endpoint.

The displacement from AA to BB can also be recovered by:

AB→=B−A.\overrightarrow{AB}=B-A.

Subtraction is directed. The vector B−AB-A points from AA to BB, while A−BA-B points back from BB to AA.

Exercise: Find an endpoint

An object starts at A=[−1,2]A=[-1,2] and moves by d=[3,−4]\mathbf{d}=[3,-4]. What is the endpoint B=A+dB=A+\mathbf{d}?

Compute it first, then check your number.

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HintCombine matching coordinates

Compute −1+3-1+3 and 2+(−4)2+(-4) separately.

SolutionAdd the displacement to the start

Coordinate-wise addition gives:

B=[−1,2]+[3,−4]=[2,−2].B=[-1,2]+[3,-4] =[2,-2].

The displacement moves three units along the positive first-axis direction and four units along the negative second-axis direction.

Exercise: Recover a displacement

A movement begins at A=[2,−1]A=[2,-1] and ends at B=[−2,4]B=[-2,4]. What is the displacement AB→\overrightarrow{AB}?

Compute it first, then check your number.

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HintSubtract start from end

Use AB→=B−A\overrightarrow{AB}=B-A and preserve the coordinate order.

SolutionKeep the direction of subtraction

The displacement from AA to BB is:

AB→=[−2,4]−[2,−1]=[−2−2, 4−(−1)]=[−4,5].\overrightarrow{AB} =[-2,4]-[2,-1] =[-2-2,\ 4-(-1)] =[-4,5].

Reversing the subtraction would produce [4,−5][4,-5], the displacement from BB back to AA.

A Displacement Can Be Translated

The vector [3,−1][3,-1] describes the same displacement wherever its arrow is drawn, provided its length and direction do not change. Starting at [1,2][1,2], it ends at [4,1][4,1]. Starting at [−2,5][-2,5], it ends at [1,4][1,4].

This freedom distinguishes a displacement vector from a position vector. A position vector uses the chosen origin as its start. A displacement only needs its own start and end points; the arrow can be translated to another location without changing the vector.

Exercise: Translate a displacement

The displacement d=[2,3]\mathbf{d}=[2,3] is drawn once from A=[0,0]A=[0,0] and again from C=[5,−1]C=[5,-1]. Which statement is correct?

Choose the correct interpretation

Select one choice, then check.

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HintCompare change rather than location

In both drawings, the endpoint is two units right and three units up from the start.

SolutionPreserve length and direction

Both arrows encode the coordinate change [2,3][2,3]. The first ends at [2,3][2,3], while the second ends at:

[5,−1]+[2,3]=[7,2].[5,-1]+[2,3]=[7,2].

Their locations differ, but their changes in each coordinate are identical.

Coordinates Depend on a Reference Frame

A position such as [3,2][3,2] is incomplete until the origin, axes, orientation, and units are known. Moving the origin changes the coordinates assigned to the same physical point. Changing the scale from metres to centimetres changes the numbers as well.

A displacement also uses axes and units, but translating the origin shifts the coordinates of both endpoints equally. Their difference remains unchanged as long as the axes and units stay the same.

This distinction appears in many settings. An embedding can be treated as a position in a representation space. A parameter update is a displacement in a parameter space. In both cases, the interpretation depends on the coordinate system in which the vector was defined.

Exercise: Distinguish location from change

Which statement describes a displacement rather than a position?

Choose the displacement statement

Select one choice, then check.

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HintLook for a start-to-end change

A position says where something is. A displacement says how its position or state changed.

SolutionRead the role of the vector

“Changed by” describes a difference between the state before and after an update, so [−0.2,0.5][-0.2,0.5] is a displacement. The statement “located at” instead assigns a position relative to an origin.

Name the Start and End

When a vector is drawn as an arrow, ask whether its start is the coordinate origin or another point. If it begins at the origin and ends at PP, it is the position vector OP→\overrightarrow{OP}. If it begins at AA and ends at BB, it is the displacement AB→=B−A\overrightarrow{AB}=B-A.

The next lesson studies vector addition, the operation that combines displacements and moves a starting point to a new endpoint.

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