Is there a way to find a perpendicular vector without using a matrix/cross product?

I need help with finding perpendicular vectors, is there a way of finding a perpendicular vector to two others without using a matrix/ cross product rule?|||Hello



Yes, you can use the scalar product: the scalar product of two perpendicular vectors = 0:



Let your two vectors be (1,2,3) and (4,5,6). And the unknown vector = (x,y,z), then set up



(x,y,z)*(1,2,3) = 0 and

(x,y,z)*(4,5,6) = 0

-------

which is written out:

x + 2y + 3z = 0 and

4x + 5y + 6z = 0



now set x = any number, like x = 1:

and solve for y and z:



(1): 1 + 2y + 3z = 0 --%26gt; *2

(2): 4 + 5y + 6z = 0

-------------------

(3): 2 + 4y + 6z = 0 --%26gt; (2-3):

2 +y = 0

y = -2 plug into (1)

1 - 4 + 3z = 0

z = 1



the vector (1, -2,1) is perpendicular to (1,2,3) and (4,5,6)



Regards|||You don't have to use a matrix. You can calculate the coefficients of the perpendicular vector directly. This formula is from wikipedia:


a 脳 b = (a2b3 鈭?a3b2) i + (a3b1 鈭?a1b3) j + (a1b2 鈭?a2b1) k


a 脳 b = (a2b3 鈭?a3b2, a3b1 鈭?a1b3, a1b2 鈭?a2b1).


where


a= a1i+a2j+a3k


b = b1i+b2j+b3k





Or if a and b lie in one of the coordinate planes, you can calculate the magnitude of the vector from:


|a| |b| sin(theta)


and then manually assign the third unit direction vector.