Determine a unit vector that is orthogonal to u = [3, -4, 1] and v = [2, 3, -4]
If w = u×v then that would be w/|w|
To determine a unit vector that is orthogonal to vectors u = [3, -4, 1] and v = [2, 3, -4], we can follow these steps:
Step 1: Find the cross product of u and v.
To find the cross product of two vectors, u and v, we can use the following formula:
u x v = [u[1]*v[2] - u[2]*v[1], u[2]*v[0] - u[0]*v[2], u[0]*v[1] - u[1]*v[0]]
Calculating the cross product of u and v, we get:
u x v = [(3*3) - (-4*(-4)), (-4*2) - (3*1), (2*(-4)) - (3*3)]
= [9 - 16, -8 - 3, -8 - 9]
= [-7, -11, -17]
Step 2: Normalize the cross product.
To find the unit vector that is orthogonal to u and v, we need to normalize the cross product vector we obtained in step 1.
The normalization of a vector can be done by dividing the vector components by its magnitude.
The magnitude of a vector [a, b, c] can be obtained using the formula:
Magnitude = sqrt(a^2 + b^2 + c^2)
In our case, the magnitude of the cross product vector [-7, -11, -17] is:
Magnitude = sqrt((-7)^2 + (-11)^2 + (-17)^2)
= sqrt(49 + 121 + 289)
= sqrt(459)
≈ 21.42 (rounded to two decimal places)
Finally, dividing each component of the cross product vector by its magnitude, we get the unit vector orthogonal to u and v:
u x v (normalized) = [-7/21.42, -11/21.42, -17/21.42]
≈ [-0.327, -0.514, -0.796]
Therefore, the unit vector orthogonal to u = [3, -4, 1] and v = [2, 3, -4] is approximately [-0.327, -0.514, -0.796].
To find a unit vector that is orthogonal (perpendicular) to two given vectors, you can use the cross product of the two vectors. The cross product will give you a vector that is orthogonal to both vectors.
1. Start by calculating the cross product of the given vectors u and v. The cross product is found by taking the determinant of a 3x3 matrix:
i j k
3 -4 1 (u)
2 3 -4 (v)
= (12 - (-12))i - (4 - (-8))j + (-12 - 6)k
= 24i + 12j - 18k
2. Now, we have a vector that is orthogonal to both u and v, which is (24, 12, -18). However, to find a unit vector, we need to normalize it.
3. To normalize the vector, calculate the magnitude (length) of the vector. The magnitude of a vector (a, b, c) is found using the formula sqrt(a^2 + b^2 + c^2):
||(24, 12, -18)|| = sqrt(24^2 + 12^2 + (-18)^2)
= sqrt(576 + 144 + 324)
= sqrt(1044)
= sqrt(4 * 261)
= 2 * sqrt(261)
4. Finally, divide each component of the vector by its magnitude to obtain a unit vector:
Unit vector = (24 / (2 * sqrt(261)), 12 / (2 * sqrt(261)), -18 / (2 * sqrt(261)))
Simplifying further:
Unit vector = (12 / sqrt(261), 6 / sqrt(261), -9 / sqrt(261))
Therefore, a unit vector that is orthogonal to u = [3, -4, 1] and v = [2, 3, -4] is approximately (0.571, 0.286, -0.429).