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Suppose that u_1 = (1, 1, 0, 0) and u_2 = (0, 0, 1, 1). Find two non-zero vectors v_1 and v_2 that are orthogonal to both u_1 and u_2 but are not parallel to each other (i.e., do not have v_1 = t v_2 for some t). Make sure you show that all of these facts are true. (This is partly why the cross-product is not well-defined in four dimensions. There\'s no single choice of a vector that is orthogonal to two others.)

Solution

This is a tricky question. It doesn\'t require any computations, just some logics

We can see that the given two vectors are u1 = <1,1,0,0> and u2 = <0,0,1,1>

Now, we need to find such two vector, whose dot product with both these vectors should be 0. This is how those two vectors will be orthogonal to the given two vectors. We also need to be careful that the two orthogonal vectors should not be mutually parallel.

Such two non-zero vectors could be:

<1,-1,0,0> and <0,0,1,-1>