Determine a vector that is orthogonal to both (2, 3, - 1) and (5, 1, 0) X

The answer to this problem is any vector (x, - 5x, - 13x). For example (1,-5,-13), (-1,5,13), (2,-10,-26), ...

Step-by-step explanation:

When are two vectors orthogonal to each other?

Two vectors are orthogonal to each other when the dot product between them is equal to zero.

Dot product.

Suppose we have two vectors, (a, b, c) and (d, e, f). The dot product between them is:

(a, b, c). (d, e, f) = ad + be = cf

Now our problem

Initially, our vector is (x, y, z) and we want to know the values of x, y, and z

Our vector is orthogonal to (2, 3, - 1), so:

(x, y, z). (2, 3, - 1) = 0

2x + 3y - z = 0

z = 2x + 3y

Our vector is also orthogonal to (5, 1, 0), so:

(x, y, z). (5, 1, 0) = 0

5x + y = 0

y = - 5x;

Since z = 2x + 3y and y = - 5x, we have that z = 2x - 15x = - 13x.

So the answer to this problem is any vector (x, - 5x, - 13x). For example (1,-5,-13), (-1,5,13), (2,-10,-26), ...