Answer
$\frac{-7}{\sqrt {19}}$, $\frac{-21}{19}i+ \frac{21}{19}j-\frac{7}{19}k$
Work Step by Step
Given: $a=3i-3j+k$ , $b=2i+4j-k$
Change the form of $a$ and $ b$.
$a=\lt3,-3,1\gt$ , $b=\lt2,4,-1\gt$
Scalar Projection $b$ onto $a$ can be calculated as follows:
$\frac{a \times b }{|a|}=\frac{(3 \times 2)+( -3 \times 4)+(1 \times -1)}{\sqrt {{(3)^{2}+(-3)^{2}}+(1)^{2}}}$
$=\frac{6-12-1}{\sqrt {19}}$
$=\frac{-7}{\sqrt {19}}$
Vector Projection $b$ onto $a$ can be calculated as follows:
$\frac{a \times b }{|a|^{2}}\times a=\frac{-7}{19}\lt3,-3,1\gt$
$=\lt\frac{-21}{19}, \frac{21}{19},\frac{-7}{19}\gt$
Change vector projection back into $i+j+k$ form.
$=\frac{-21}{19}i+ \frac{21}{19}j+\frac{-7}{19}k$
$=\frac{-21}{19}i+ \frac{21}{19}j-\frac{7}{19}k$
Hence,
Scalar Projection $b$ onto $a$ = $\frac{-7}{\sqrt {19}}$,
Vector Projection $b$ onto $a$=$\frac{-21}{19}i+ \frac{21}{19}j-\frac{7}{19}k$
