这是在最好（最快）方式来计算两个向量垂直于所述第三矢量（X）和也彼此垂直？

What is the best (fastest) way to compute two vectors that are perpendicular to the third vector(X) and also perpendicular to each other?

这是我应该如何计算这个向量将现在：

This is how am I computing this vectors right now:

// HELPER - unit vector that is NOT parallel to X
x_axis = normalize(X);
y_axis = crossProduct(x_axis, HELPER);
z_axis = crossProduct(x_axis, y_axis);

我知道有解决这一无限多的，我不关心哪一个会是我的解决办法。

I know there is infinite number of solutions to this, and I don't care which one will be my solution.

这背后的问题：我需要构造变换矩阵，在那里我知道哪个方向应该X轴（矩阵第一列）被人指指点点。我需要计算Y和Z轴（第二和第三列）。正如我们所知，所有轴必须垂直于对方。

What is behind this question: I need to construct transformation matrix, where I know which direction should X axis (first column in matrix) be pointing. I need to calculate Y and Z axis (second and third column). As we know, all axes must be perpendicular to each other.

推荐答案

我做了什么，只要 X＆LT;＆GT; 0 或 Y'LT ;＆GT; 0 是

然后规范化载体。

[ X,Y,Z]·[-Y,X,0] = -X*Y+Y*X = 0
[ X,Y,Z]·[-X*Z,-Y*Z,X*X+Y*Y] = -X*X*Z-Y*Y*Z+Z*(X*X+Y*Y) = 0
[-Y,X,0]·[-X*Z,-Y*Z,X*X+Y*Y] = Y*X*Z+X*Y*Z = 0

这就是所谓的向量的零空间的。

如果 X = 0 和 Y = 0 然后 A = [1,0 ，0] ， B = [0,1,0] 。