Multiplication of Vectors and Scalars

Multiplication of Vectors and Scalars

If we decided to walk three times as far on the first leg of the trip considered in the preceding example, then we would walk \(\text{3 }×\text{ 27}\text{.}\text{5 m}\), or 82.5 m, in a direction \(\text{66}\text{.}0\text{º}\) north of east. This is an example of multiplying a vector by a positive scalar. Notice that the magnitude changes, but the direction stays the same.

If the scalar is negative, then multiplying a vector by it changes the vector’s magnitude and gives the new vector the opposite direction. For example, if you multiply by –2, the magnitude doubles but the direction changes. We can summarize these rules in the following way: When vector \(\mathbf{A}\) is multiplied by a scalar \(c\),

• the magnitude of the vector becomes the absolute value of \(c\)\(A\),
• if \(c\) is positive, the direction of the vector does not change,
• if \(c\) is negative, the direction is reversed.

In our case, \(c=3\) and \(A=27.5 m\). Vectors are multiplied by scalars in many situations. Note that division is the inverse of multiplication. For example, dividing by 2 is the same as multiplying by the value (1/2). The rules for multiplication of vectors by scalars are the same for division; simply treat the divisor as a scalar between 0 and 1.