Answer
$\frac{-37}{7}$, $\lt\frac{74}{49}, \frac{-111}{49},\frac{222}{49}\gt$
Work Step by Step
Given: $a=\lt-2,3,-6\gt$ , $b=\lt5,-1,4\gt$
Scalar Projection $b$ onto $a$ can be calculated as follows:
$\frac{a \times b }{|a|}=\frac{(-2 \times 5)+( 3 \times -1)+(-6 \times 4)}{\sqrt {{(-2)^{2}+(3)^{2}}+(-6)^{2}}}$
$=\frac{-10-3-24}{\sqrt {49}}$
$=\frac{-37}{7}$
Vector Projection $b$ onto $a$ can be calculated as follows:
$\frac{a \times b }{|a|^{2}}\times a=\frac{-37}{49}\lt-2,3,-6\gt$
$=\lt\frac{74}{49}, \frac{-111}{49},\frac{222}{49}\gt$
Hence,
Scalar Projection $b$ onto $a$ = $\frac{-37}{7}$,
Vector Projection $b$ onto $a$=$\lt\frac{74}{49}, \frac{-111}{49},\frac{222}{49}\gt$
