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What does vector multiplication achieve?
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Sometimes there is a need to change the length of a vector or change its length and reverse its direction. This is accomplished by a type of multiplication in which vectors are multiplied by scalars. As an example, the product $2 \mathrm{v}$ denotes the vector that has the same direction as $\mathbf{v}$ but twice the length, and the product $-2 \mathbf{v}$ denotes the vector that is oppositely directed to $\mathbf{v}$ and has twice the length.
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