Find a unit vector perpendicular to both A = (2,-3,1) and B = (1,2,-4).
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)).
To find a unit vector perpendicular to both A = (2,-3,1) and B = (1,2,-4), we can use the cross product.
The cross product of two vectors is a vector that is perpendicular to both of the original vectors. The magnitude of the resulting vector is equal to the product of the magnitudes of the original vectors times the sine of the angle between them.
To find the cross product of A and B, we can use the following formula:
C = A x B
Let's calculate the cross product step by step:
1. Calculate the components of the cross product using the formula:
C = (Ay * Bz - Az * By, Az * Bx - Ax * Bz, Ax * By - Ay * Bx)
Given A = (2,-3,1) and B = (1,2,-4), we have:
Cx = Ay * Bz - Az * By = (-3 * -4) - (1 * 2) = 12 - 2 = 10
Cy = Az * Bx - Ax * Bz = (1 * 1) - (2 * -4) = 1 + 8 = 9
Cz = Ax * By - Ay * Bx = (2 * 2) - (-3 * 1) = 4 + 3 = 7
So, C = (10, 9, 7).
2. Calculate the magnitude of C:
|C| = sqrt(Cx^2 + Cy^2 + Cz^2) = sqrt(10^2 + 9^2 + 7^2) = sqrt(100 + 81 + 49) = sqrt(230)
3. Calculate the unit vector perpendicular to A and B:
To get a unit vector, we divide each component of C by its magnitude:
unit vector = (Cx/|C|, Cy/|C|, Cz/|C|)
unit vector = (10/sqrt(230), 9/sqrt(230), 7/sqrt(230))
Therefore, a unit vector perpendicular to both A and B is approximately (0.667, 0.60, 0.467).