Question

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Do unit vectors have dimensions? Why are unit vectors used? Can unit vectors be multiplied by scalars? Explain your responses to all three questions.

Answer 1

Unit vectors are said to be dimensionless because they have been normalized.

Unit vectors are normally used to indicate a certain direction by the vector, not it's magnitude. The magnitude of all unit vectors is 1. Let's say we have a certain unit vector that gives us a direction. If we multiply that by a scalar quantity, say force (magnitude), then we can find the force vector along that direction. We can also see the values of the force along each component of that vector.
Ex: Let's say I have the vector: \(\vec{V}=<1, 2>\). To normalize this, I have to take the square root of the sum of each component squared. Mathematically, this means \(R=\sqrt{1^2+2^2}=\sqrt{5}\). Therefore, the UNIT vector is now:
\[\vec{U}=<\frac{ 1 }{ \sqrt{5} },\frac{ 2 }{ \sqrt{5} }>\]This is referred to as normalizing a vector.
Now let's say that I know the value of the force along that direction. Therefore, the force vector can be expressed as\[\large\vec{F}=\text{F} \vec{U}=\text{F}<\frac{ 1 }{ \sqrt{5} },\frac{ 2 }{ \sqrt{5} }>=<\frac{ \text{F} }{ \sqrt{5} },\frac{ 2 \text{F} }{ \sqrt{5} }>\]Now we know the components of the force along that direction.