The Difference Between the Lines 𝐀 + t𝐁 and 𝐁 + t(𝐀 − 𝐁)

The Difference Between the Lines 𝐀 + t𝐁 and 𝐁 + t(𝐀 − 𝐁)

A line in vector form is defined by two components: a base point that determines its position, and a direction vector that determines its orientation. Two expressions may involve the same vectors but still represent completely different lines when either the base point or the direction vector changes. The expressions

• L₁: 𝐀 + t𝐁
• L₂: 𝐁 + t(𝐀 − 𝐁)

provide a clear example of how distinct lines arise from different vector components.

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1. Line L₁: 𝐀 + t𝐁

The expression 𝐀 + t𝐁 describes a line passing through the point represented by vector 𝐀 with direction vector 𝐁. As the real parameter t varies, the expression generates all points on the line.

• Base point: 𝐀
• Direction vector: 𝐁

This is the line through 𝐀 directed along 𝐁.

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2. Line L₂: 𝐁 + t(𝐀 − 𝐁)

The expression 𝐁 + t(𝐀 − 𝐁) describes a different line. Its base point is 𝐁, and its direction vector is the difference vector 𝐀 − 𝐁, which points from 𝐁 toward 𝐀. This vector determines the orientation of the line.

• Base point: 𝐁
• Direction vector: 𝐀 − 𝐁

This is the line through 𝐁 in the direction of 𝐀 − 𝐁.

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3. Why the Lines Are Distinct

Although both expressions involve the vectors 𝐀 and 𝐁, they define different lines because their essential components differ:

• The base points are not the same (𝐀 versus 𝐁).
• The direction vectors are not the same (𝐁 versus 𝐀 − 𝐁).

A change in either the base point or the direction vector produces a new geometric line. Changing both components necessarily yields a different line except in special degenerate cases (such as both direction vectors being scalar multiples of one another and both base points lying on the resulting line).

Thus:

• L₁ begins at 𝐀 and extends along the direction of 𝐁.
• L₂ begins at 𝐁 and extends toward 𝐀 along 𝐀 − 𝐁.

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4. Structural Interpretation

A vector equation of the form P + tD always separates cleanly into:

• P — the point fixing the position of the line
• D — the vector fixing its orientation

The expressions 𝐀 + t𝐁 and 𝐁 + t(𝐀 − 𝐁) differ in both components, so they define two distinct geometric lines. Understanding how base points and direction vectors determine a line is fundamental in vector geometry and in the study of parametric and affine representations of lines.

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Conclusion

The vector equations 𝐀 + t𝐁 and 𝐁 + t(𝐀 − 𝐁) describe different lines because they use different base points and different direction vectors. Recognising this structure is essential for analysing and manipulating parametric lines within vector calculus, geometry, and linear algebra.