Operations involving vectors

There are three kinds of values that appear in vector algebra: scalars, points, and vectors. We can try to add these, subtract them, and multiply them in various ways. Sometimes these operations make sense, and sometimes they don't, with results as in the following table:
 Add: ScalarPointVector Subtract:ScalarPointVector Multiply: ScalarPointVector
Scalar  Scalar   Scalar   Scalar Vector
Point  Point  Vector Point 
Vector  Point Vector  Vector  Vector Depends
There are various ways to multiply two vectors, with results as in the table below:
NameSymbolResult Order matters?Depends on lengths? Depends on right-hand rule?
Dot product ScalarNo YesNo
Cross product× Scalar in 2D,
vector in 3D
YesYes Yes
Tensor product DyadYes NoNo
We don't cover the tensor product (or dyads) in this course, but I put it in there just to emphasize that there are more operations on vectors than the ones that we do cover.
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