Usually we use dimensional analysis to find the dimension of acceleration or force, but can we do the same thing to find the dimension of the vector acceleration, and the vector force, or we can't because those vectors haven't dimension?
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2$\begingroup$ vectors do have dimensions, v.g. the force vector has dimension of force. $\endgroup$– ZeroTheHeroCommented Aug 19, 2018 at 21:37
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3$\begingroup$ You might be getting confused by the fact that a vector is sometimes thought of as a unit vector in a particular direction, multiplied by a scalar magnitude. It's conventional for the magnitude to carry the vector's dimensions, and the unit vector to be considered dimensionless. $\endgroup$– Philip WoodCommented Aug 19, 2018 at 22:52
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1 Answer
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Vectors can have units. With common vectors like acceleration, velocity, and displacement, the unit of the whole vector is the same as each of its components. That is, a displacement vector of [3 m, 4 m, 5 m] will have a unit of meters. A force vector of [10 N, 5 N] has a unit of newtons. This can help with calculations since, for example, an acceleration vector times mass should yield a force vector.
One way to check that this works is to confirm that the magnitude of a vector has the correct units. The magnitude of a force vector had better have units of newtons. To use the previous example: \begin{align} \left\|\vec{F}\right\| &= \left\|[10 N, 5 N]\right\| \\ &= \sqrt{(10 N)^2 + (5 N)^2} \\ &= \sqrt{125 N^2} \\ &\approx 11.2 N \end{align} as expected.
In other areas of physics, vector components can have different units. In optical ray tracing, the components of a ray vector are a distance and an angle. So, the vector does not have an overall unit. One consequence is that, unlike the force vector example above, the magnitude of a ray vector is meaningless.
