Question

What can be said about the vectors \(\mathbf{u}\) and \(\mathbf{v}\) if (a) the projection of \(\mathbf{u}\) onto \(\mathbf{v}\) equals \(\mathbf{u}\) and \((b)\) the projection of \(\mathbf{u}\) onto \(\mathbf{v}\) equals \(\mathbf{0}\) ?

Short answer

In context (a), the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are in the same direction. In context (b), the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are perpendicular to each other.

Step-by-step solution

Analyze condition (a)

Under condition (a), the projection of \(\mathbf{u}\) onto \(\mathbf{v}\) equals \(\mathbf{u}\). This means the vector \(\mathbf{u}\) lies along \(\mathbf{v}\) as the projection is itself. Thus, \(\mathbf{u}\) and \(\mathbf{v}\) are in the same direction.

Analyze condition (b)

In situation (b), the projection of \(\mathbf{u}\) onto \(\mathbf{v}\) equals \(\mathbf{0}\). This is only possible when the angle between the vectors \(\mathbf{u}\) and \(\mathbf{v}\) is 90 degrees, meaning \(\mathbf{u}\) and \(\mathbf{v}\) are perpendicular.

Draw Conclusions

From step 1 and step 2, we can conclude that in situation (a), \(\mathbf{u}\) and \(\mathbf{v}\) are in the same direction, and in situation (b), \(\mathbf{u}\) and \(\mathbf{v}\) are perpendicular to each other.