Scalar & Vector Projections


A projection is like a shadow casted on a vector. The definition of scalar and vector projections are shown below.

Scalar Projection

of \(\vec{a}\text{ on }\vec{b}=\vert\text{proj}\left(\vec{a}\text{ on }\vec{b}\right)\vert= \dfrac{\vec{a}\cdot\vec{b}}{\vert\vec{b}\vert} \)

of \(\vec{b}\text{ on }\vec{a}=\vert\text{proj}\left(\vec{b}\text{ on }\vec{a}\right)\vert=\dfrac{\vec{a}\cdot\vec{b}}{\vert\vec{a}\vert} \)

Vector Projection

Vector projection on b will result in a vector in the direction of \(\vec{b} \).

proj\(\left(\vec{a}\text{ on }\vec{b}\right)=\dfrac{\vec{a}\cdot\vec{b}}{\vert\vec{b}\vert} \hat{b} \)

\(=\left(\dfrac{\vec{a}\cdot\vec{b}}{\vec{b}\cdot\vec{b}}\right)\vec{b}=\dfrac{\vec{a}\cdot\vec{b}}{\vert\vec{b}\vert^2}\vec{b} \)

Vector Projection Calculator



Enter in two vectors below. Then, Click the + button to toggle between addition and subtraction.









Find the scalar and vector projections of \(\vec{u}\text{ onto }\vec{v}\text{, if }\vec{u}=(5, 6, -3)\text{ and }\vec{v}=(1, 4, 5) \).


Find the scalar and vector projections of \(\vec{v}\text{ onto }\vec{u} \).


If \(\vec{u}\text{ and }\vec{v} \) are non-zero vectors, but Proj \(\vec{u}\text{ onto }\vec{v}=\vec{0} \), what conclusion can be drawn?


If \(\vec{u}\text{ and }\vec{v} \) are non-zero vectors, does it follow that Proj \(\vec{u}\text{ onto }\vec{v}=\vec{0} \)?


Find the projection of \(\vec{PQ} \) onto each of the coordinate axes, where \(P \) is the point (2, 3, 5) and \(Q \) is the point (-1, 2, 5).




©MyHomeworkRewards 2016 - 2024