Editing Computational geometry
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To every vector except <math>\mathbf{v}</math> (except <math>\mathbf{v} = \mathbf{0}</math>), we can assign a vector <math>\hat{\mathbf{v}}</math> that has the same direction but a length of exactly 1. This is called a ''unit vector''. The unit vector can be calculated as follows: | To every vector except <math>\mathbf{v}</math> (except <math>\mathbf{v} = \mathbf{0}</math>), we can assign a vector <math>\hat{\mathbf{v}}</math> that has the same direction but a length of exactly 1. This is called a ''unit vector''. The unit vector can be calculated as follows: | ||
:<math>\hat{\mathbf{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|}</math> | :<math>\hat{\mathbf{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|}</math> | ||
− | There are two ''elementary unit vectors'': [1,0], denoted <math>\hat{\ | + | There are two ''elementary unit vectors'': [1,0], denoted <math>\hat{\mathbf{i}}</math>, and [0,1], denoted <math>\hat{\mathbf{j}}</math>. That is, the unit vectors pointing along the positive x- and y-axes. Note that we can write a vector <math>\mathbf{v}</math> as <math>v_x\hat{\mathbf{i}} + v_y\hat{\mathbf{j}}</math>. |
− | + | ||
==Obtaining a vector of a given length in the same direction as a given vector== | ==Obtaining a vector of a given length in the same direction as a given vector== | ||
Suppose we want a vector of length <math>l</math> pointing in the same direction as <math>\mathbf{v}</math>. Then, all we need to do is scale <math>\mathbf{v}</math> by the factor <math>\frac{l}{\|\mathbf{v}\|}</math>. Thus our new vector is | Suppose we want a vector of length <math>l</math> pointing in the same direction as <math>\mathbf{v}</math>. Then, all we need to do is scale <math>\mathbf{v}</math> by the factor <math>\frac{l}{\|\mathbf{v}\|}</math>. Thus our new vector is |