What is the projection of $< 2, - 7, 1 >$ $<2,-7,1 >$ onto $< 4, - 5, 9 >$ $<4,-5,9 >$ ?

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Answer 1 · 2018-04-22T12:01:10

The projection is $= \frac{7}{\sqrt{122}} < 4, - 5, 9 >$ $=7/sqrt122<4, -5, 9>$

The projection of $\to v$ $vecv$ onto $\to u$ $vecu$ is

$p r o j_{\to u} (\to v) = \frac{< \to u, \to v >}{< \to u, \to u >} \to u$ $proj_(vecu)(vecv)= (< vecu, vecv >)/ (< vecu, vecu >) vecu$

$\to u = < 4, - 5, 9 >$ $vecu = <4, -5, 9>$

$\to v = < 2, - 7, 1 >$ $vecv= <2, -7,1>$

The dot product is

$< \to u, \to v > = < 4, - 5, 9 > . < 2, - 7, 1 >$ $< vecu, vecv > = <4, -5, 9> .<2, -7,1>$

$= (4 \times 2) + (- 5 \times 2) + (9 \times 1)$ $=(4xx2)+(-5xx2)+(9xx1)$

$= 8 - 10 + 9$ $=8-10+9$

$= 7$ $=7$

The magnitude of $\to u$ $vecu$ is

$< \to u, \to u > = | | < 4, - 5, 9 > | | = \sqrt{4^{2} + {(- 5)}^{2} + 9^{2}}$ $< vecu, vecu > = ||<4, -5, 9>|| =sqrt(4^2+(-5)^2+9^2)$

$= \sqrt{16 + 25 + 81}$ $=sqrt(16+25+81)$

$= \sqrt{122}$ $=sqrt122$

Therefore,

$p r o j_{\to u} (\to v) = \frac{7}{\sqrt{122}} < 4, - 5, 9 >$ $proj_(vecu)(vecv)=7/sqrt122<4, -5, 9>$

What is the projection of <2,-7,1 ><2,−7,1><2,-7,1 > onto <4,-5,9 ><4,−5,9><4,-5,9 >?