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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 λ, 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 λ:
|λ A|=λ |A| if λ >0
Let us now consider multiplication by negative numbers. Suppose two points P and P’ on a plane are at the same distance from the origin O, but in opposite directions as seen from O, i.e. P’OP is a straight line. It is clear that vector OP’ is the negative of vector OP, i.e. OP’ = (- 1) ×OP. Thus multiplication of a vector by –1 merely reverses its direction without changing its magnitude. It follows that the multiplication of a vector by –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 λgives a vector λA whose direction is opposite to that of A but whose magnitude is (−λ) times |A|:
|λA| = -λ|A| = |λ| |A|; if λ< 0
To summarise, the multiplication of a vector by a real number either leaves the direction unchanged (if the number is positive) or reverses the direction (if the number is negative); while the magnitude of the vector is scaled by the magnitude of the number.

The number λ need not be a pure number (such as 2, 3, -1, -2, etc.) without any dimensions. In fact λ 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 λ. Then the dimensions of the product λA will be the product of the dimensions of λ 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 × T = L).

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