A vector is a geometric object which has both magnitude (i.e. length) and direction. A vector is generally represented by a line segment with a certain direction connecting the initial point A and the terminal point B as shown in the figure below and is denoted by 
Projection of a Vector on another vector
The projection of a vector 
onto another vector 
is given as
Computing vector projection onto another vector in Python:
# import numpy to perform operations on vectorimport numpy as np u = np.array([1, 2, 3]) # vector uv = np.array([5, 6, 2]) # vector v: # Task: Project vector u on vector v # finding norm of the vector vv_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
The projection of a vector onto a plane is calculated by subtracting the component of which is orthogonal to the plane from .
where, 
is the plane normal vector.
Computing vector projection onto a Plane in Python:
# import numpy to perform operations on vectorimport numpy as np # vector u u = np.array([2, 5, 8]) # vector n: n is orthogonal vector to Plane Pn = 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 Pprint("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 -0.64705882]
