Find a non-zero vector orthogonal to (1, 2, -1)
One possible non-zero vector that is orthogonal to (1, 2, -1) is (2, -1, -2).
To find a non-zero vector orthogonal to a given vector, we can use the concept of the dot product.
The dot product of two vectors is given by the sum of the products of their corresponding components. For two vectors to be orthogonal, their dot product must be zero.
Let's find a vector orthogonal to (1, 2, -1):
Let's assume the orthogonal vector is (x, y, z).
The dot product of the two vectors will be:
(1, 2, -1) · (x, y, z) = 1*x + 2*y + (-1)*z = x + 2y - z
Since the vectors are orthogonal, their dot product must be zero:
x + 2y - z = 0
Now, we can choose any values for x, y, and z that satisfy this equation. Let's choose x = 2 and y = 1:
2 + 2(1) - z = 0
2 + 2 - z = 0
4 - z = 0
z = 4
So, a non-zero vector orthogonal to (1, 2, -1) is (2, 1, 4).