Projection of a Vector on another vector
Computing vector projection onto another vector in Python:
# import numpy to perform operations on vector import numpy as np u = np.array([1, 2, 3]) # vector u v = np.array([5, 6, 2]) # vector v: # Task: Project vector u on vector v # finding norm of the vector v v_norm = np. sqrt(sum(v**2)) # Apply the formula as mentioned above # for projecting a vector onto another vector # find dot product using np.dot() proj_of_u_on_v = (np.dot(u, v)/v_norm**2)*v print("Projection of Vector u on Vector v is: ", proj_of_u_on_v)
Output:
Projection of Vector u on Vector v is: [1.76923077 2.12307692 0.70769231]
One liner code for projecting a vector onto another vector:
(np.dot(u, v)/np.dot(v, v))*v
Projection of a Vector onto a Plane
Computing vector projection onto a Plane in Python:
# import numpy to perform operations on vector import numpy as np # vector u u = np.array([2, 5, 8]) # vector n: n is orthogonal vector to Plane P n = np.array([1, 1, 7]) # Task: Project vector u on Plane P # finding norm of the vector n n_norm = np. sqrt(sum(n**2)) # Apply the formula as mentioned above # for projecting a vector onto the orthogonal vector n # find dot product using np.dot() proj_of_u_on_n = (np.dot(u, n)/n_norm**2)*n # subtract proj_of_u_on_n from u: # this is the projection of u on Plane P print("Projection of Vector u on Plane P is: ", u - proj_of_u_on_n)
Output:
Projection of Vector u on Plane P is: [ 0.76470588 3.76470588 -0.64705]
How to use Python to calculate Vector Projection?
Question from StackOverFlow
Is there an easier command to compute vector projection? I am instead using the following:
x = np.array([ 3, -4, 0])
y = np.array([10, 5, -6])
z=float(np.dot(x, y))
z1=float(np.dot(x, x))
z2=np. sqrt(z1)
z3=(z/z2**2)
x*z3
Answer:
Maybe this is what you really want:
np.dot(x, y) / np.linalg.norm(y)
This should give the projection of vector x
onto vector y
- see https://en.wikipedia.org/wiki/Vector_projection. Alternatively, if you want to compute the projection of y
onto x
, then replace y
with x
in the denominator (norm
) of the above equation.
EDIT: As @VaidAbhishek commented, the above formula is for the scalar projection. To obtain vector projection multiply scalar projection by a unit vector in the direction of the vector onto which the first vector is projected. The formula then can be modified as:
y * np.dot(x, y) / np.dot(y, y)
for the vector projection of x
onto y
.