How to Find a Vector Projection

Vector Projection

• 1

  Calculate the magnitude of vector v, supposing that you want to find the vector projection of a vector u on the vector v. For example, if u = 2i + j and v = -3i + 4j, the magnitude of v is given by sqrt( (-3)^2 + 4^2) = 5.

• 2

  Compute the scalar product of the two vectors. This example produces 2 * -3 + 1 * 4 = -2.

• 3

  Use the formula "projvu = ((u * v) / |v|^2) * v to obtain the projected vector. In the example above, the final value is -2 / 25 * (-3, 4) = (6/25, -8/25) = 6/25i -- 8/25j.

Scalar Projection

• 4

  Compute the magnitude of vector v, supposing that you want to find the scalar projection of a vector u on the vector v. Let u = -3i + 5j and v = -7i --j. The magnitude of v will then be sqrt((-7)^2 + (-1)^2) = sqrt(50).

• 5

  Compute the scalar product of the two vectors. In this case, the result is (-3) * (-7) + 5 * (-1) = 21 -- 5 = 16.

• 6

  Use the formula "sprojvu = (u * v )/ |v| to compute the scalar projection of vector u over v. This produces 16 / sqrt(50) = 8 * sqrt(2) / 5.