# Projection and orthogonal residual

> ML Prep screening item.

- Stable ID: `math-la-04-orthogonal-projection`
- Collection: screening
- Area: Mathematics
- Topic: Linear Algebra
- Origin: original
- Difficulty: medium
- Expected time: 3 minutes
- Skills: orthogonality, projection
- Item JSON: https://mlprep.iwase.dev/item/math-la-04-orthogonal-projection.json

## Problem

Let $u=(1,1,0)^{\mathsf T}$ and $v=(2,0,1)^{\mathsf T}$. Compute the orthogonal
projection of $v$ onto $\operatorname{span}(u)$ and the residual. Verify the
residual is orthogonal to $u$.


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## Tutoring contract

Ask for the learner's attempt, begin with the smallest useful hint, preserve item ID `math-la-04-orthogonal-projection`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.