# Vector Solutions, Shortcuts, Formulas

## Position Vector

A vector $\vec{AB}$ simply means the displacement from a point A to the point B. Now consider a situation that a girl moves from A to B and then from B to C (Fig 10.7). The net displacement made by the girl from point A to the point C, is given by the vector $\vec{AC}$ and express $\vec{AC}=\vec{AB}+\vec{AC}$
This is known as the triangle law of vector addition.

If we have two vectors $\vec{a}\: and\: \vec{b}$ by the two adjacent sides of a parallelogram in magnitude and direction (Fig 10.9), then their sum is represented in magnitude and direction by the diagonal of the parallelogram through their common point. This is known as the parallelogram law of vector addition.
Using triangle law , $\vec{OA}+\vec{AC}=\vec{OC}$
$\vec{OA}+\vec{OB}=\vec{OC}$ (since $\vec{AC}=\vec{OB}$ Which is parallelogram law. Thus, we may say that the two laws of vector addition are equivalent to each other

1) For any two vector $\vec{a}\: and\: \vec{b}$
$\vec{a}+\vec{b}=\vec{b}+\vec{a}$ (commutative properties)

2) For any three properties $\vec{a},\vec{b}\:and\: \vec{c}$
$(\vec{a}+\vec{b})+\vec{c}=\vec{a}+(\vec{b}+\vec{c})$ (Associative properties )

Note: The associative property of vector addition enables us to write the sum of three vectors $\vec{a},\vec{b,\vec{c}}\: as\:\vec{a}+\vec{b}+\vec{c}$ as without using brackets. Note that for any vector $\vec{a}$ , we have $\vec{a}+\vec{0}=\vec{0}+\vec{a}=\vec{a}$
Here , the zero vector is called additive identity for the vector addition .

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