# Multiplication of Vectors by Real Numbers

When a vector is multiplied by a real number, the result is another vector. We will illustrate this by taking the example of position vectors. Let us first consider the multiplication of vectors by positive real numbers. Suppose two vectors**OP**and

**OQ**are in the same direction (i.e. OPQ is a straight line) but the length of vector

**OQ**is twice that of vector

**OP**. We say that

OQ = 2OP
Thus the multiplication of OP by 2 has doubled its length but its direction remains unchanged. This means that if a vector A is multiplied by a real positive number Î», the result is a vector, which is written as ∈ A whose direction is the same as that of A but whose magnitude is changed by a factor Î»:

**|**Î»

**A|=**Î»

**|A|**if Î» >0

**OPâ€™**is the negative of vector

**OP**, i.e.

**OPâ€™**= (- 1) Ã—

**OP**. Thus multiplication of a vector by â€“1 merely reverses its direction without changing its magnitude. It follows that the multiplication of a vector by â€“2 reverses its direction and also doubles its magnitude.

Thus the general rule is that the multiplication of a vector A by a negative number Î»gives a vector Î»A whose direction is opposite to that of A but whose magnitude is (âˆ’Î») times |A|:

**A| = -**Î»

**|A| = |**Î»

**| |A|;**if Î»< 0

The number Î» need not be a pure number (such as 2, 3, -1, -2, etc.) without any dimensions. In fact Î» could be a scalar having its own dimensions and A could be a physical quantity having its own dimensions which could, be different from those of Î». Then the dimensions of the product Î»A will be the product of the dimensions of Î» and A. For example, if we multiply a velocity vector (dimensions LT-1) by a time interval (which is a scalar of dimension T), we get a displacement vector (dimension LT-1 Ã— T = L).