Vector Algebra: Orthogonal Projection?

📌 CONCEPT: The orthogonal projection of a vector α onto a vector μ is zero if and only if μ is orthogonal to α.

📐 RULE / FORMULA: To find the magnitude of μ, we can use the formula |μ| = √(μ · μ), where μ · μ is the dot product of μ with itself.

💡 WORKED EXAMPLE: Let α = <2, 3, -1> and assume μ = <a, b, c>. Since μ is orthogonal to α, we have (a*2 + b*3 + c*(-1)) = 0. We are also given that |α| = 4 units. Using the formula |μ| = √(a^2 + b^2 + c^2), we can find the possible range of the magnitude of μ.

⚠️ COMMON MISTAKE: Students often confuse orthogonality with perpendicularity, and may incorrectly assume that the magnitude of μ is directly proportional to the magnitude of α.