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Mathematics · University

Scalar coefficient in an orthogonal projection

The coefficient multiplying u differs from the scalar projection onto the unit vector u/||u||.

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Method and conditions

The coefficient multiplying u differs from the scalar projection onto the unit vector u/||u||.

projᵤ(v) = [(v·u)/(u·u)]u

Worked examples

Let v = (6,-9) and u = (1,8). Write the orthogonal projection of v onto span(u) as k u. Find k. Round the final result to 4 decimal places if needed.

Use k = (v·u)/(u·u) = (-66)/(65).

-1.0154

Let v = (7,5) and u = (7,2). Write the orthogonal projection of v onto span(u) as k u. Find k. Round the final result to 4 decimal places if needed.

Use k = (v·u)/(u·u) = (59)/(53).

1.1132

Try it

  1. Let v = (5,-1) and u = (3,7). Write the orthogonal projection of v onto span(u) as k u. Find k. Round the final result to 4 decimal places if needed.

  2. Let v = (-1,9) and u = (3,1). Write the orthogonal projection of v onto span(u) as k u. Find k. Round the final result to 4 decimal places if needed.

Check your work

Answers and marking notes
  1. 0.1379

    Use k = (v·u)/(u·u) = (8)/(58).

  2. 0.6

    Use k = (v·u)/(u·u) = (6)/(10).

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