Vector Projection

$\vec{a}\downarrow \vec{b}$ Vector Projection

The idea of a vector casting a shadow on another vector and turning that ‘shadow’ into a vector. The projection vector is the vector $\overrightarrow{ON}$

Formula

Vector projection of $\vec{a}\downarrow \vec{b}$ is: $\overrightarrow{ON}=\frac{\vec{a}\cdot \vec{b}}{|\vec{b}|}(\hat{b})$ Vector projection of $\vec{b}\downarrow \vec{a}$ is: $\overrightarrow{ON}=\frac{\vec{a}\cdot \vec{b}}{|\vec{a}|}(\hat{a})$

Scalar Projection

The scalar projection is the magnitude of a vector projection on another vector.

Formula

Scalar projection of $\vec{a}\downarrow \vec{b}$ is: $ON=|\vec{a}|\cos \theta$ $ON=\frac{\vec{a}\cdot \vec{b}}{|\vec{b}|}$ Scalar projection of $\vec{b}\downarrow \vec{a}$ is: $ON=|\vec{b}|\cos \theta$ $ON=\frac{\vec{a}\cdot \vec{b}}{|\vec{a}|}$

Polarity

• $ON>0$ if $\theta <90$
• $ON=0$ if $\theta =90$
• $ON<0$ if $\theta >90$