How do you find the value of α in v1 = αv2 + e that minimizes the error vector?

The value of α that minimizes the length of the error vector e is found using the vector projection formula: α = (v1 · v2) / (v2 · v2). This ensures that the error vector e is orthogonal to v2, giving the minimum possible length.

What is the answer to GATE EC 2023 Question 10 on vector projection?

The answer is 2/7 (Option C). Using the projection formula α = (v1 · v2) / (v2 · v2), the dot products of the given vectors yield α = 2/7.

Why must the error vector e be orthogonal to v2 for minimum length?

When e is orthogonal to v2, any change in α only increases the length of e. The shortest possible error is always the perpendicular component of v1 with respect to v2, which is why the projection formula gives the optimal α.

What is the vector projection formula?

The projection of vector v1 onto v2 is given by: proj = [(v1 · v2) / (v2 · v2)] × v2. The scalar coefficient α = (v1 · v2) / (v2 · v2) is the value that minimizes the error vector in v1 = αv2 + e.

Vector Projection: Minimize Error Vector e in v1 = αv2 + e

EC > GATE 2023 > Linear Algebra

Let

be two vectors. The value of the coefficient α in the expression v₁ = αv₂ + e which minimizes the length of the error vector e, is

7/2

-2/7

2/7

-7/2

Correct : C

The correct answer is Option C: 2/7.
To minimize the length of the error vector e = v₁ - αv₂, the error must be orthogonal to v₂. This is the core idea of vector projection.
Setting e ⊥ v₂ means e · v₂ = 0, which directly gives the projection formula:
α = (v₁ · v₂) / (v₂ · v₂)
Substituting the dot products of the given vectors yields α = 2/7.
Intuitively, αv₂ is the component of v₁ along the direction of v₂, and e is what remains - the perpendicular part. The perpendicular distance is always the shortest possible, which is why this value of α gives the minimum error.

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