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.