26.4 Scalar Multiplication of a Vector
A vector can be multiplied by a scalar (i.e a constant) such as \(\, 2\underline {a}, \quad 3\underline {b}, \quad -3\underline {a}, \quad 5\underline {b}\)
In general the vector \(\lambda \underline {a}\) has magnitude \(\left |\lambda \right |\left |\underline {a}\right |\), and its direction is the same as that of \(\underline {a}\) when \(\lambda >0\) and opposite when \(\lambda <0\).
Note 26.2. The absolute value matters. Multiplying by \(-3\) does not give a vector of “magnitude \(-3\left |\underline {a}\right |\)” — a magnitude is a length and cannot be negative. It gives a vector three times as long pointing the other way, which is what \(-3\underline {a}\) in the list above means.
This is also what the unit vector construction relies on: dividing \(\underline {a}\) by the positive number \(\left |\underline {a}\right |\) leaves the direction alone and scales the length to exactly \(1\).
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.