Questions: Assume the length of u is 2. On your scrap work: 1. First, draw the vectors obtained from u, v via the Gram-Schmidt process.

Solution Steps

• Given vectors: \( \mathbf{u} \) and \( \mathbf{v} \)
• Length of \( \mathbf{u} \) is 2

• The Gram-Schmidt process will generate an orthogonal set of vectors from \( \mathbf{u} \) and \( \mathbf{v} \).

1. Let \( \mathbf{u}_1 = \mathbf{u} \)
2. Compute the projection of \( \mathbf{v} \) onto \( \mathbf{u}_1 \): \[ \text{proj}_{\mathbf{u}_1} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{u}_1}{\mathbf{u}_1 \cdot \mathbf{u}_1} \mathbf{u}_1 \]
3. Subtract this projection from \( \mathbf{v} \) to get the orthogonal vector \( \mathbf{u}_2 \): \[ \mathbf{u}_2 = \mathbf{v} - \text{proj}_{\mathbf{u}_1} \mathbf{v} \]

• Normalize \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \) to get an orthonormal set of vectors.

1. Normalize \( \mathbf{u}_1 \): \[ \mathbf{e}_1 = \frac{\mathbf{u}_1}{\|\mathbf{u}_1\|} \] Given \( \|\mathbf{u}_1\| = 2 \), we have: \[ \mathbf{e}_1 = \frac{\mathbf{u}}{2} \]

2. Normalize \( \mathbf{u}_2 \): \[ \mathbf{e}_2 = \frac{\mathbf{u}_2}{\|\mathbf{u}_2\|} \]

Final Answer