I think that's all they were looking for.neelakash said:Homework Statement
Let A be an arbitrary vector and e be a unit vector in some fixed direction.Show that
A=e(A.e)+e x (A x e)
What is the geometrical significance of each of the two terms?
Homework Equations
The Attempt at a Solution
I can show it easily.As the first term (a dot product) is the component in the e direction and the 2nd term(a cross product) is the component in the perpendicular direction.
What else geometrical significance they may be talking about?
The geometric significance of this equation is that it represents the vector projection of vector A onto the plane perpendicular to vector e. This is also known as the vector rejection of A from e.
The vector projection is calculated by taking the dot product of A with the normalized vector e (A.e), multiplying it by e, and adding the cross product of A and e (A x e). This results in a vector that is perpendicular to e and has the same magnitude as the projection of A onto the plane.
The scalar term A.e represents the magnitude of the projection of A onto e. It is equivalent to the length of the shadow that A would cast if a light source were placed perpendicular to e.
The cross product term (A x e) represents the direction of the vector projection. It ensures that the resulting vector is perpendicular to e, and thus lies on the plane perpendicular to e.
Yes, this equation can be used for vectors in three-dimensional space. However, the vector projection will be onto the plane perpendicular to e, rather than just a line as it is in two-dimensional space.