Question

Projection of a vector onto itself is

• A. equal to $0$.
• B. equal to $1$.
• C. equal to $-1$.
• D. equal to its magnitude.

Answer 1

The correct option is D equal to its magnitude.

Let a vector be $\overrightarrow{A}$
We know that the projection of $\overrightarrow{A}\text{along}\overrightarrow{A}$ is given as
$\frac{\overrightarrow{A}\cdot \overrightarrow{A}}{|\overrightarrow{A}|}=\frac{|\overrightarrow{A}||\overrightarrow{A}|\cos \theta }{|\overrightarrow{A}|}=|\overrightarrow{A}|\cos \theta$
Since, the projection is on itself,
$\cos \theta =\cos 0^{\circ }=1$
$\Rightarrow |\overrightarrow{A}|\cos \theta =|\overrightarrow{A}|$
Hence, projection of a vector onto itself is its magnitude.