Final answer:
To create two orthogonal 1000-vectors v and w with no equal entries, one must ensure their dot product equals zero. By constructing vectors with single, isolated non-zero elements, such as (1, 0, 0, ..., 0) for v and (0, 1, 0, ..., 0) for w, orthogonality is achieved. Alternatively, another valid pair could be v = (3, 0, 4, ..., 0) and w = (-4, 0, 3, ..., 0), as their dot product is also zero.
Step-by-step explanation:
Finding Orthogonal Vectors
To find two orthogonal 1000-vectors v and w with no entries equal, we must ensure that the dot product of these vectors equals zero. An easy way to construct such vectors is to have one vector with a single nonzero entry and the other vector with a nonzero entry in a different position.
For example, vector v could be (1, 0, 0, ..., 0) and vector w could be (0, 1, 0, ..., 0). The dot product v ∙ w = (1∙0) + (0∙1) + (0∙0) + ... + (0∙0) = 0, which confirms that v and w are orthogonal.
Another pair of orthogonal vectors might be v = (3, 0, 4, 0, ..., 0) and w = (-4, 0, 3, 0, ..., 0). The dot product v ∙ w = (3∙-4) + (0∙0) + (4∙3) = -12 + 12 = 0, again showing that the vectors are orthogonal.