Vector Projections 1F265D

1. **Problem Statement:** Determine which of the given projection relationships among vectors $A$, $B$, $C$, $D$, and $E$ holds true. 2. **Recall the projection formula:** The projection of vector $\mathbf{u}$ onto vector $\mathbf{v}$ is given by: $$\mathrm{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}$$ This means the projection is a vector parallel to $\mathbf{v}$ scaled by the scalar projection. 3. **Analyze each given relationship:** - (A) $\mathrm{proj}_B A = B$ This means projecting $A$ onto $B$ results exactly in $B$. For this to be true, $A$ must have a component along $B$ equal in magnitude and direction to $B$ itself. - (B) $\mathrm{proj}_C B = A$ Projecting $B$ onto $C$ equals $A$. Since $C$ is a short vector near the origin and $A$ is longer and at upper right, this is unlikely. - (C) $\mathrm{proj}_E A = D$ Projecting $A$ onto $E$ equals $D$. $E$ is at lower right, $D$ at upper left, so projection onto $E$ cannot produce $D$ which points in a different direction. - (D) $\mathrm{proj}_D A = E$ Projecting $A$ onto $D$ equals $E$. $D$ is upper left, $E$ is lower right, opposite directions, so this is unlikely. 4. **From the graph description:** - $B$ is on positive y-axis (top center). - $A$ is upper right. - $C$ is short near origin pointing slightly up-right. - $D$ is upper left. - $E$ is lower right. 5. **Check (A) in detail:** Since $B$ is vertical and $A$ is upper right, projecting $A$ onto $B$ will give a vector along $B$ but shorter than $B$ unless $A$ is exactly equal to $B$ in direction and magnitude. 6. **Conclusion:** Only (A) $\mathrm{proj}_B A = B$ is plausible if $A$ has the same component along $B$ as $B$ itself. **Final answer:** The relationship that holds is (A) $\mathrm{proj}_B A = B$.