Final answer:
To expand the vector v in terms of the vectors u1, u2, and u3, first check if the vectors are mutually orthonormal by computing their dot products and verifying that they are unit vectors. Then, express v as a linear combination of the orthonormal vectors by projecting v onto each vector, and multiply this by the corresponding basis vector.
Step-by-step explanation:
To find the expansion of the vector v=⟨−0,0,2⟩ in terms of the vectors u1 = 1/6 ⟨1,−2,1⟩, u2 =1/11⟨−3,−1,1⟩, and u3 = 1/66 (−1,−4,−7), we need to check if these vectors are mutually orthonormal. Two vectors are orthonormal if they are orthogonal (their dot product is zero) and normalized (each vector has a magnitude of one).
First, calculate the dot products of the vectors u1, u2, and u3 to check for orthogonality and normalize them to check if each is a unit vector. Then, if these vectors form an orthonormal basis for the space, you can express vector v as a linear combination of u1, u2, and u3. This is done by projecting v onto each of these basis vectors using the dot product and multiplying by the appropriate basis vector.
The expansion coefficients are found by taking the dot product of v with each ui and dividing by the magnitude squared of ui (since they are not unit vectors).
The expansion, provided the u vectors are indeed orthonormal, will be:
v = (v · u1)u1 + (v · u2) u2 + (v · u3) u3