Multiplication of a Vector by a Scalar
Consider a vector \(\vec a\). What happens if you multiply this vector by 2? What will the vector \(2\vec a\) represent?
Intuitively, the answer should be clear: the new vector will be twice the length of the original vector, but in the same direction:
The vector \(\frac{1}{2}\vec a\) will be a vector in the same direction as \(\vec a\), but with a length equal to half of the length of \(\vec a\):
We have seen how to interpret the vector \(  \vec a\), given the vector \(\vec a\):
The two vectors are in opposite directions.
What about the vector \(  2\vec a\)? The direction of this vector will be opposite to the direction of \(\vec a\), and its length will be twice the length of \(\vec a\):
The vector \(  \frac{1}{2}\vec a\) will have a direction opposite to that of \(\vec a\), and will have a length equal to half the length of \(\vec a\):
To summarize:

Multiplying a vector by a positive scalar (real number) preserves its direction, and scales its length by the magnitude of the scalar.

Multiplying a vector by a negative scalar reverses its direction, and scales its length by the magnitude of the scalar.

It the magnitude of the scalar is greater than 1, then the new vector is longer than the original vector; if it is less than 1, then the new vector is shorter; if the scalar is equal to 1, the new vector has the same length as the original.