# 2.2 Multiplication by a scalar

In the collection of vectors sketched in Section 2.1, although **v** is not equal to **c**, the vectors **v** and **c** are closely related: **c** is a vector in the same direction as **v**, but it is twice as long as **v**. Thus it is natural to write 2**v** for **c**, since we can think of a journey represented by **c** as being a journey **v** followed by a second journey **v**.

In an analogous way, we can write for **e**, the vector whose magnitude is times that of **f** and whose direction is that of −**f**.

## Scalar multiple of a vector

Let *k* be a scalar and **v** a vector. Then *k***v** is the vector whose magnitude is |*k*| times the magnitude of **v**, that is, ‖*k***v**‖ = |*k*| ‖**v**‖, and whose direction is

the direction of

vifk> 0,

the direction of −

vifk< 0.

If *k* = 0, then *k***v** = **0**.

## Example 19

For each of the vectors shown below, decide whether it is a multiple of any of the other vectors; if it is, write down an equation of the form **v**_{1} = *k***v**_{2} that specifies the relationship between them.

### Answer

The vector **d** is in the same direction as **a**, but none of the other vectors is; also, the length of **d** is two-thirds that of **a**. Hence

Next, **e** is along the same line as **b** but in the opposite direction; none of the others is along the same line. Also, the length of **e** is three times that of **b**. Hence

Finally, **c** and **f** are not multiples of any of the other vectors.

## Example 20

For the vector **d** in Exercise 19, sketch 3**d** and −2**d**.

### Answer

The vector 3**d** is in the same direction as **d**, but its magnitude is three times that of **d**; the vector −2**d** is in the direction opposite to that of **d**, and its magnitude is twice that of **d**.