Find a unit vector perpendicular to both A = (2,-3,1) and B = (1,2,-4).

Answer 1

To find a vector that is perpendicular to both A and B, we can take the cross product of the two vectors. The cross product of two vectors gives us a vector that is perpendicular to both of them.

The cross product of A = (2, -3, 1) and B = (1, 2, -4) is given by the formula:

A x B = (AyBz - AzBy, AzBx - AxBz, AxBy - AyBx)

Plugging in the values:

A x B = (-3*(-4) - 1*2, 1*1 - 2*(-4), 2*2 - (-3)*1)
= (-10, 9, 7)

To turn this into a unit vector, we divide each component of the vector by its magnitude:

|A x B| = sqrt((-10)^2 + 9^2 + 7^2)
= sqrt(100 + 81 + 49)
= sqrt(230)

To find the vector perpendicular to both A and B, we divide each component of the cross product vector by its magnitude:

A_perp_B = (-10/sqrt(230), 9/sqrt(230), 7/sqrt(230))

So, a unit vector perpendicular to both A = (2, -3, 1) and B = (1, 2, -4) is A_perp_B = (-10/sqrt(230), 9/sqrt(230), 7/sqrt(230)).