Equivalent Vectors

A vector is defined by two things only: its magnitude (length) and its direction. It does not matter where a vector starts. Two vectors are called equivalent vectors if they have the same magnitude and point in the same direction, even if their starting points (called initial points) are completely different.

This is one of the most important ideas in vector geometry: you can slide a vector anywhere in the plane without changing what vector it represents, as long as you do not change its length or the direction it points.

The easiest way to test equivalence is to write each vector in component form, as \( \langle a, b \rangle \). Two vectors \( \vec{u} = \langle a_1, b_1 \rangle \) and \( \vec{v} = \langle a_2, b_2 \rangle \) are equivalent exactly when

\( a_1 = a_2 \) and \( b_1 = b_2 \)

In other words, the horizontal change and vertical change must both match. If a vector runs from point \( A(1, 2) \) to point \( B(4, 6) \), its component form is

\( \vec{AB} = \langle 4 - 1,\ 6 - 2 \rangle = \langle 3, 4 \rangle \)

Any other vector with components \( \langle 3, 4 \rangle \), no matter where it starts, is equivalent to \( \vec{AB} \).

You can also compare vectors using their magnitude and direction. The magnitude of \( \vec{u} = \langle a, b \rangle \) is

\( |\vec{u}| = \sqrt{a^2 + b^2} \)

and its direction is described by its direction angle. Two vectors are equivalent when both their magnitudes are equal and their direction angles are equal. This confirms the same fact from a different angle (literally): matching components always produce matching magnitude and direction, and vice versa.

Students often confuse equivalent vectors with parallel vectors. Parallel vectors point in the same direction (or exactly opposite directions), but they can have different lengths. Equivalent vectors are a stricter case: they must be parallel and have the exact same magnitude and the exact same direction (not the opposite one). So every pair of equivalent vectors is parallel, but not every pair of parallel vectors is equivalent.

Determine whether \( \vec{u} \) from \( A(-2, 1) \) to \( B(2, 4) \) is equivalent to \( \vec{v} \) from \( C(0, 0) \) to \( D(4, 3) \).

Step 1: Find the components of \( \vec{u} \).

\( \vec{u} = \langle 2 - (-2),\ 4 - 1 \rangle = \langle 4, 3 \rangle \)

Step 2: Find the components of \( \vec{v} \).

\( \vec{v} = \langle 4 - 0,\ 3 - 0 \rangle = \langle 4, 3 \rangle \)

Step 3: Compare. Since \( \langle 4, 3 \rangle = \langle 4, 3 \rangle \), the vectors have the same components, so \( \vec{u} = \vec{v} \). They are equivalent vectors, even though they start at different points.

Recognizing equivalent vectors lets you move a vector freely when solving geometry and physics problems, add vectors tip-to-tail, and simplify vector diagrams without losing information. It is also a foundation for special vectors such as the unit vector, and for the operations you will use throughout vector algebra.

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