Dot Product of Vectors
Vector Algebra: Scalar Product Formulas
1. Definition & Coordinate Form
| Form | Formula |
|---|---|
| Definition (Using Angle $$ \theta $$) |
$$ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos\theta $$ |
| Coordinate Form (If $$ \vec{a}=a_1\hat{i} + a_2\hat{j} + a_3\hat{k} $$) |
$$ \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 $$ |
| Angle between Vectors | $$ \cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|} $$ |
2. Important Properties
Perpendicularity ($$ \perp $$)
If $$ \vec{a} \perp \vec{b} $$, then:
$$ \vec{a} \cdot \vec{b} = 0 $$
If $$ \vec{a} \perp \vec{b} $$, then:
$$ \vec{a} \cdot \vec{b} = 0 $$
Parallelism ($$ \parallel $$)
If $$ \theta = 0 $$: $$ \vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| $$
If $$ \theta = \pi $$: $$ \vec{a} \cdot \vec{b} = -|\vec{a}||\vec{b}| $$
If $$ \theta = 0 $$: $$ \vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| $$
If $$ \theta = \pi $$: $$ \vec{a} \cdot \vec{b} = -|\vec{a}||\vec{b}| $$
Self Product
$$ \vec{a} \cdot \vec{a} = |\vec{a}|^2 $$
$$ \vec{a} \cdot \vec{a} = |\vec{a}|^2 $$
| Commutative Property | $$ \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} $$ |
| Distributive Property | $$ \vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} $$ |
| Scalar Multiple | $$ (\lambda \vec{a}) \cdot \vec{b} = \lambda (\vec{a} \cdot \vec{b}) $$ |
3. Projection of a Vector
| Projection Type | Formula |
|---|---|
| Projection of $$ \vec{a} $$ on $$ \vec{b} $$ | $$ \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \quad \text{or} \quad \vec{a} \cdot \hat{b} $$ |
| Projection of $$ \vec{b} $$ on $$ \vec{a} $$ | $$ \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|} \quad \text{or} \quad \vec{b} \cdot \hat{a} $$ |
| Projection Vector (Vector component of $$ \vec{a} $$ along $$ \vec{b} $$) |
$$ \left( \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \right) \hat{b} = \left( \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \right) \vec{b} $$ |