Change the elements of this column matrix: (3, 4)T so that the vector it represents is twice as long and remains pointing in the same direction.

### A good answer might be:

You could carefully work out the correct answer using what you already know. Or you could take a guess and see if it works:   2 times (3, 4)T   =   (6, 8)T

# Scaling

Does the guess work?

• (New length)2 = 62 + 82   =   2232 + 2242   =
22(32+42)   =   22(old length)2
• So, taking roots, (New length) = 2(old length)
• By similar triangles, the direction is the same.

Or you could notice: new direction   =   arc tan( 8/6 )   =   arc tan( 4/3 )   =   old direction. It looks like the guess worked.

In talking about vectors, a real number is sometimes called a scalar.

Scaling a geometrical vector means keeping its orientation the same but changing its length by a scale factor. It is like changing the scale of a picture; the distances expand or shrink, but the directions remain the same.

If a vector is represented by a column matrix (x, y)T then scaling by the a number multiplies each element:

```a(x, y)T  =  (ax, ay)T
```

This works for 3 dimensional vectors also: If v is represented by (x, y, z)T then av is represented by (ax, ay, az)T.

### QUESTION 2:

What is 0.5*(36.4, -18.9)T ?