Parallel Vectors

What Are Parallel Vectors?

A vector is a quantity that has both magnitude and direction.

Vectors are parallel if they have the same direction or opposite direction.

Two non-zero vectors, u and v, are parallel if and only if one is a scalar multiple of the other.
Mathematically, vectors u and v are parallel if:

u = kv

where k is a scalar (non-zero real number).

If k > 0, the vectors point in the same direction.
If k < 0, the vectors point in opposite directions (sometimes called anti-parallel).

Both components of one vector must be in the same ratio to the corresponding components of the parallel vector.

Example:

How To Define Parallel Vectors?

Two vectors are parallel if they are scalar multiples of one another.

If u and v are two non-zero vectors and u = cv, then u and v are parallel.

The following diagram shows several vectors that are parallel.

How To Determine If The Given 3-Dimensional Vectors Are Parallel?

Example: Determine which vectors are parallel to v = <-3, -2, 5>

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What Are The Conditions For Two Lines To Be Parallel Given Their Vector Equations?

Lines are parallel if the direction vectors are in the same ratio.

Example: If the lines l₁: \(r = \left( {\begin{array}{*{20}{c}}1\\{ - 5}\\7\end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}{a - 1}\\{ - a - 1}\\b\end{array}} \right)\) and l₂: \(r = \left( {\begin{array}{*{20}{c}}9\\3\\{ - 8}\end{array}} \right) + \mu \left( {\begin{array}{*{20}{c}}{2a}\\{3 - 5a}\\{15}\end{array}} \right)\).

Find the values of a and b.

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What Is A Vector Is, How To Add And How To Prove Vectors Are Parallel And Collinear?

Examples:

1. A, B, C are midpoints of their respective lines. Find the vector OB.
2. N = midpoint of OB, M = midpoint of OA. Show that MN is parallel to AB.
3. Given the vectors, prove that the three given points are collinear.

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How To Answer A Question That Involves Vectors - Lines, Parallel, Perpendicular & Intersection?

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