School of Mathematics and Statistics, The University of Sydney
 4. Multiplication of a vector by a scalar
Main menu Section menu   Previous section Next section button disabled When are vectors parallel?
Glossary
Examples
Quiz

The geometric meaning of multiplication by a scalar

Page 1 of 3 

Using a head-to-tail procedure, a vector v can be added to itself to give the vector v + v.

C v B v A 2v

We have v = ---> AB and v = ----> BC. It’s natural to write ---> AB + ----> BC = - --> AC as 2v. The vector 2v has length 2|v| and is in the same direction as v. The notion of multiplication of a vector by a positive integer is then generalised to define the vector sv for all scalars s, as follows.

The vector sv is called a scalar multiple of the vector v and is defined to be the vector of length |s||v|, in the same direction as v if s > 0 and in the opposite direction if s < 0. If s = 0 then sv is the zero vector.

For example, -3v has three times the magnitude of v but points in the opposite direction; 1 2v (also written v- 2) has magnitude |v| 2 and has the same direction as v.

v v - 3v 1-v 2

Note that -1v has the same magnitude as v but has the opposite direction, and so is the same vector as the negative of v, that is, -1v = -v.

Feedback
Main menu Section menu   Previous section Next section button disabled When are vectors parallel?