finding a perpendicular vector using the concept of dot product

finding a perpendicular vector using the concept of dot product
find a vector perpendicular to both [ -1, 3, 4 ] and [-2, -1, 3]

let [x ,y, z] be the required vector

since [x ,y, z] is perpendicular to both [ -1, 3, 4 ] and [-2, -1, 3]
-1x + 3y + 4z = 0
-2x -1y + 3z = 0

three unknowns and two equations
choose one unknown as arbitrary

put x = a

3y + 4z = a
-y +3z = 2a

solving

y = -5a / 13
z = 7a / 13

so if a = 13

x = 13 , y = -5 , z = 7

therefore [13 , -5 , 7] or any of it's scalar multiples will be the required vector
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