Question

Construct an orthonormal basis for vectors {(1,2,3),(2,3,4),(2,5,7)}

Answer 1

To construct an orthonormal basis for the given vectors using the Gram-Schmidt process, you generalize each vector sequentially, ensuring each is orthogonal to the previous, and then normalizing them to have unit length.

Step-by-step explanation:

To construct an orthonormal basis for the given set of vectors {(1,2,3),(2,3,4),(2,5,7)}, we use the Gram-Schmidt process followed by normalization. The process converts a set of vectors into an orthogonal set, and then we turn it into an orthonormal set by dividing each vector by its magnitude.

1. Start with the first vector (1,2,3) and normalize it by dividing by its magnitude, getting the first unit vector u1.
2. Next, subtract the projection of the second vector (2,3,4) along u1 from itself to get a vector orthogonal to u1. Normalize this to get the second unit vector u2.
3. Finally, for the third vector (2,5,7), subtract the projections along u1 and u2 from it, and normalize the result to get the third unit vector u3.

This set {u1, u2, u3} will be an orthonormal set because all vectors are mutually orthogonal and of unit length, which means their scalar products will vanish, indicating angles of 90° between them, just as the Cartesian coordinate system dictates.