Distribution Property of Dot Product

Question

Since the dot product has the property that for three vectors $a,b,c$

$a \cdot (b+c) = a \cdot b + a \cdot c$

Is that also true for $(a+b) \cdot (a+b) = a\cdot a + 2a\cdot b + b\cdot b$

?

Thank you

Answer 1

Assuming that you also know the fact that the dot product is commutative, then you can indeed prove the equation you give: \begin{align} (a+b)\cdot(a+b) &= (a+b)\cdot a + (a+b)\cdot b \\ &= a\cdot(a+b) + b\cdot(a+b) \\ &= a\cdot a + a\cdot b + b\cdot a + b\cdot b \\ &= a\cdot a + a\cdot b + a\cdot b + b\cdot b \\ &= a\cdot a + 2a\cdot b + b\cdot b. \end{align}

Answer 2

Yes its true because

$b.a=a.b$ as it is completely scalar.

But distributive property doesnt hold true in case of cross product.