Question

If the projection of \(\mathbf{u}\) onto \(\mathbf{v}\) has the same magnitude as the projection of \(\mathbf{v}\) onto \(\mathbf{u}\), can you conclude that \(\|\mathbf{u}\|=\|\mathbf{v}\|\) ? Explain.

Short answer

No, if the projection of vector \( \mathbf{u} \) onto vector \( \mathbf{v} \) has the same magnitude as the projection of vector \( \mathbf{v} \) onto vector \( \mathbf{u} \), it does not necessarily mean that vectors \( \mathbf{u} \) and \( \mathbf{v} \) have the same magnitude.

Step-by-step solution

Understanding the projection operation

Recall that the projection of vector \( \mathbf{u} \) onto vector \( \mathbf{v} \) is given by \[ \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \mathbf{v} \]. Similarly the projection of \( \mathbf{v} \) onto \( \mathbf{u} \) is given by \[ \frac{\mathbf{v} \cdot \mathbf{u}}{\|\mathbf{u}\|^2} \mathbf{u} \].

Equating magnitudes of projections

\[ \left\|\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \mathbf{v}\right\| = \left\|\frac{\mathbf{v} \cdot \mathbf{u}}{\|\mathbf{u}\|^2} \mathbf{u}\right\| \] The magnitudes of vectors are always positive, so we can write \[ \left|\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2}\right| \|\mathbf{v}\| = \left|\frac{\mathbf{v} \cdot \mathbf{u}}{\|\mathbf{u}\|^2}\right| \|\mathbf{u}\| \] Given that \(\mathbf{u} \cdot \mathbf{v}\) and \(\mathbf{v} \cdot \mathbf{u}\) are equivalent, and |a/b| = |a|/|b|. We can simplify this equation further into \[ \left|\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|}\right| = \left|\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|}\right| \]

Evaluate the relationship between magnitudes of vectors

It is clearly seen that the magnitudes of vector \(\mathbf{u}\) and vector \(\mathbf{v}\) can be different and still satisfy the equation of projection magnitudes. Hence, if the projection of \(\mathbf{u}\) unto \(\mathbf{v}\) and projection of \(\mathbf{v}\) unto \(\mathbf{u}\) have the same magnitude, we can’t conclude that \( \|\mathbf{u}\| = \|\mathbf{v}\| \).