Given any (nonzero) vectors [color=#000][math]\vec v[/math][/color] and [color=#000][math]\vec w[/math][/color], you can always split [color=#000][math]\vec v[/math][/color] into the sum of two vectors [color=#ff0000][math]\vec v_1[/math][/color] and [color=#0000ff][math]\vec v_2[/math][/color], with [color=#ff0000][math]\vec v_1[/math][/color] in the same direction as [color=#000][math]\vec w[/math][/color] and [color=#0000ff][math]\vec v_2[/math][/color] orthogonal to [color=#000][math]\vec w[/math][/color].[br][br]In the illustration below, you are free to drag the tips of the vectors [color=#000][math]\vec v[/math][/color] and [color=#000][math]\vec w[/math][/color].

To see how you can use [color=#000][math]\vec v[/math][/color] and [color=#000][math]\vec w[/math][/color] to find [color=#ff0000][math]\vec v_1[/math][/color] and [color=#0000ff][math]\vec v_2[/math][/color], think about the following:[br][br][list][br][*] [math]\vec v[/math] should be the sum of [math]\vec v_1[/math] and [math]\vec v_2[/math], so [math]\vec v_1+\vec v_2=\vec v[/math].[br][*] [math]\vec v_1[/math] should be parallel to [math]\vec w[/math], so [math]\vec v_1=k\vec w[/math] for some number [math]k[/math].[br][*] [math]\vec v_2[/math] should be orthogonal to [math]\vec w[/math], so [math]\vec v_2\,\cdot\,\vec w=0[/math].[br][/list][br][br]You can use these three equations to figure out what the number [math]k[/math] must be, and once you know [math]k[/math], you can easily find [math]\vec v_1[/math] and [math]\vec v_2[/math].