Heisenberg's Uncertainty Principle gives a relationship between the standard deviation of an object's position and its momentum.
[tex]\Delta p \cdot \Delta x = h / (4 \pi)[/tex] where
By definition, the momentum of the electron equals the product of its mass and velocity.
[tex]p = m\cdot v[/tex]
Assuming that measurement of the mass of the electron [tex]m[/tex] is accurate. It is assumed to be a coefficient of constant value. The standard deviation in the electron's velocity is thus directly related to that of its mass. That is:
[tex]\Delta p = m \cdot \Delta v[/tex]
[tex]\Delta v = 0.01 \times 10^{6} \;\text{m}\cdot \text{s}^{-1}[/tex] from the question;
[tex]\Delta p = m\cdot v \\ \phantom{\Delta p} = 0.01 \times 10^{6} \; \text{m} \cdot \text{s}^{-1} \times 9.11 \times 10^{-31} \; \text{kg}\\\phantom{\Delta p} = 9.11 \times 10^{-27} \; \text{kg} \cdot \text{m}\cdot \text{s}^{-1}[/tex]
Convert the unit of the Planck's constant to base SI units (kg, m, s, etc.) if it was provided in derived units such as joules. Doing so would allow for a dimension analysis on the accuracy of the result.
[tex]h = 6.63 \times 10^{-34} \; \text{J} \cdot \text{s}\\\phantom{h} = 6.63 \times 10^{-34} \; (\text{N}\cdot \text{m}) \cdot \text{s} \\\phantom{h} = 6.63 \times 10^{-34} \; ((\text{kg} \cdot \text{m}\cdot \text{s}^{-2}) \cdot \text{m}) \cdot \text{s}\\\phantom{h} = 6.63 \times 10^{-34} \; \text{kg} \cdot \text{m}^{2} \cdot \text{s}^{-1}[/tex]
Apply the Uncertainty Principle:
[tex]\Delta x = h/ (4 \pi \cdot \Delta p)\\\phantom{\Delta x} = 6.63 \times 10^{-34} \; \text{kg} \cdot \text{m}^{2} \cdot \text{s}^{-1} / (4 \pi \cdot 9.11\times 10^{-27} \; \text{kg} \cdot \text{m}\cdot \text{s}^{-1})\\\phantom{\Delta x} = 5.79 \times 10^{-9} \; \text{m}[/tex].
Dimensional analysis:
[tex]\Delta x[/tex] resembles the standard deviation of a position measurement. It is expected to have a unit of meter, which is the same as that of position.
