Question

Find a unit vector in R 3 that satisfies in turn each of the following conditions.

(i) The unit vector is parallel to → xy,
where x = (4, 1, −1) and y = (3, 2, 1).
(ii) The unit vector is ort

Answer 1

To find a unit vector parallel to x and y, we can find the cross product of x and y to get a vector perpendicular to both x and y, and then divide this vector by its magnitude.

Step-by-step explanation:

To find a unit vector parallel to the vectors x = (4, 1, −1) and y = (3, 2, 1), we can first find the cross product of x and y, which will give us a vector perpendicular to both x and y. Then, we can divide this vector by its magnitude to get a unit vector.

The cross product of x and y is given by x cross y = (4, 1, -1) cross (3, 2, 1).

By using the formula for the cross product and simplifying the calculation, we can find that x cross y = (3, -7, 5).

The magnitude of x cross y is square root of (3^2 + (-7)^2 + 5^2) = square root of 83.

Now, we can divide x cross y by its magnitude to get the unit vector: (3, -7, 5) / square root of 83.