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Vectors
Equal Vectors
Vector Multiplication
Vector Geometry
In these lessons, we will learn how to determine if the given vectors are parallel.
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:
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.
Example: Determine which vectors are parallel to v = <-3, -2, 5>
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.
Examples:
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