Scale a normalized 2D vector always to the same length

Question

For any normalized 2D vector, except for ( 0, 0 ), how would I scale the vector to always be the same length?

For example:

int length = 10;

vector v = vector( 0.1, 0.5 );
vector v2 = vector( 0.3, 0.8 );

// Scale v to be length of 10
// Scale v2 to be length of 10

Answer 1

A normalized vector, by definition, has length 1. To scale it to length 10, you need to multiply it by 10.

(To multiply a vector by a number, multiply every component of the vector by the number. Depending on the language you use, just writing vector = vector * number may do that automatically, or there may at least be a library function or method to do that.)

By the way, the vectors in your example code are not normalized — their length is not 1. To normalize them, do as Gajoo suggests, i.e. divide them by their length. Again, your language or vector library may already provide a function to do that.

Answer 2

You mean like this?

length = sqrt(v.x * v.x + v.y * v.y);

// normalize vector
v.x /= length;
v.y /= length;

// increase vector size
v.x *= 10
v.y *= 10

Answer 3

Normalised vectors have the same length by definition: one. If you want to scale them to an arbitrary length L instead, multiply them (or their components) by L. Note that the examples you have provided are not normalised vectors. To normalise, divide the vectors by their magnitude.

For example (component-wise, as I do not know the extent of vector algebra support in your environment):

int targetlength = 10;
vector v = vector( 0.1, 0.5 );
float vMagnitude = sqrt(v.x*v.x + v.y*v.y);
v.x = targetlength * v.x / vMagnitude;
v.y = targetlength * v.y / vMagnitude;

// v = ( 1.96, 9.81 )

Or with vector algebra:

int targetlength = 10;
vector v = vector( 0.1, 0.5 );
v = targetlength * v / v.length;