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Let $\overrightarrow{w} = c \overrightarrow{v}$ then what is $a\overrightarrow{w}$?

So is it simply:

$a(c\overrightarrow{v}) = ac\overrightarrow{v}$?

2 Answers 2

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If a is some constant value then yes. Otherwise if it is vector then its dot product with $\vec w$

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    What about $a(c\overrightarrow{v} + d\overrightarrow{w})$?? – 2017-01-29
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    still then it will be true till we remember that $a,c,d$ are constants belonging to $\mathbb{R}$ or field – 2017-01-29
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    its $ac \vec v + ad \vec w$. – 2017-01-29
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Yes you are correct! (Provided you take $c,a\in\Bbb F$, the underlying field). It follows from the definition of a Vector Space.

If $V$ is a vector space with an underlying field $\Bbb F$ then it must satisfy the Axiomatic Definition of a Vector Space. One of the Axioms is as follows,

For $x\in V$ and $c,d\in \Bbb F$ $\implies$ $ (cd)x = c(dx)$