Question

Two vectors $x$ and $y$ are shown in the figure. The projection vector of $x$ on $y$ is

• A. $\dfrac{x^{T}y}{y^{T}y}y$
• B. $x \times y$
• C. $\dfrac{x \times y}{y^{T}y}$
• D. $\dfrac{x^{T}y}{x^{T}x}x$

Hint:

Projection always lies along the direction of the vector you're projecting onto; cross products never appear in projection formulas.

Answer 1

The Correct Option is A

The projection of a vector $x$ onto another vector $y$ is given by the formula:
\[ \text{proj}_y(x) = \left( \frac{x \cdot y}{\|y\|^2} \right) y \]
In matrix notation, the dot product $x \cdot y$ is written as $x^T y$, and the squared norm of $y$ is $y^T y$.
Thus the projection vector is:
\[ \frac{x^T y}{y^T y} y \]
Option (B) is incorrect because $x \times y$ gives a vector perpendicular to both $x$ and $y$ and has nothing to do with projection.
Option (C) still contains a cross product and is therefore incorrect.
Option (D) is the projection of $y$ onto $x$, not $x$ onto $y$.
Hence, the only correct formula is option (A).
Final Answer: $\dfrac{x^{T}y}{y^{T}y}y$