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Subject:  Linear Algebra

Project (3, 4) onto (1, 0)

Find the vector projection and the perpendicular component.

Answer proj_b(a) = (3, 0)

Solution

  1. Vectors a = (3, 4), b = (1, 0)
  2. Dot product a · b = 3
  3. Magnitude squared |b|² = 1
  4. Scalar projection comp_b(a) = (a · b)/|b| = 3
  5. Coefficient (a · b)/|b|² = 3
  6. Projection onto b proj_b(a) = (3, 0)
  7. Perpendicular component a − proj_b(a) = (0, 4)

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