A projection of a vector
u onto
v is written as proj
v u. It is named as the vector projection of
u onto
v.
To find proj
v u, we use the formula below:
Take note that
\frac{\mathbf{v}}{|\mathbf{v}|} "\frac{\mathbf{u}\cdot \mathbf{v}}{|\mathbf{v}|^2}\mathbf{v}=\left ( \frac{\mathbf{u}\cdot \mathbf{v}}{|\mathbf{v}|} \right )\frac{\mathbf{v}}{|\mathbf{v}|}")
where the scalar component of
u in the direction of
v is
and the direction is
. The scalar component tells us about the magnitude of the vector proj
v u and whether it's in the same direction of
v or opposite to it. When the scalar component is positive, proj
v u is in the same direction with
v; when it is negative, proj
v u is opposite to the direction of
v.
Try to move point A below to the opposite direction of the blue vector,
v. You will see that proj
v u is opposite to the direction of
v.
Betty, Created with
GeoGebra and Thomas Calculus, Pearson.
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