Answer
$\frac{1}{9}$, $\lt\frac{4}{81}, \frac{7}{81},\frac{-4}{81}\gt$
Work Step by Step
Given: $a=\lt4,7,-4\gt$ , $b=\lt3,-1,1\gt$
Scalar Projection $b$ onto $a$ can be calculated as follows:
$\frac{a \times b }{|a|}=\frac{(4 \times 3)+( 7 \times -1)+(-4 \times 1)}{\sqrt {{(4)^{2}+(7)^{2}}+(-4)^{2}}}$
$=\frac{12-7-4}{\sqrt {81}}$
$=\frac{1}{9}$
Vector Projection $b$ onto $a$ can be calculated as follows:
$\frac{a \times b }{|a|^{2}}\times a=\frac{1}{81}\lt4,7-4\gt$
$=\lt\frac{4}{81}, \frac{7}{81},\frac{-4}{81}\gt$
Hence,
Scalar Projection $b$ onto $a$ = $\frac{1}{9}$,
Vector Projection $b$ onto $a$=$\lt\frac{4}{81}, \frac{7}{81},\frac{-4}{81}\gt$
