What is the projection of $< 3, 1, 5 >$ $<3,1,5>$ onto $< 2, 3, 1 >$ $<2,3,1>$ ?

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What is the projection of $< 3, 1, 5 >$ $<3,1,5>$ onto $< 2, 3, 1 >$ $<2,3,1>$ ?

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Answer 1 · 2018-01-20T06:45:32

The vector projection is $= < 2, 3, 1 >$ $= <2, 3, 1>$

Explanation:

The vector projection of $\to b$ $vecb$ onto $\to a$ $veca$ is

$p r o j_{\to a} \to b = \frac{\to a . \to b}{{(∣ ∣ ∣ ∣ \to a ∣ ∣ ∣ ∣)}^{2}} \to a$ $proj_(veca)vecb=(veca.vecb)/(||veca||)^2veca$

$\to a = < 2, 3, 1 >$ $veca=<2,3,1>$

$\to b = < 3, 1, 5 >$ $vecb= <3, 1,5>$

The dot product is

$\to a . \to b = < 3, 1, 5 > . < 2, 3, 1 >$ $veca.vecb =<3,1,5>. <2,3,1>$

$= (3) \cdot (2) + (1) \cdot (3) + (5) \cdot (1) = 6 + 3 + 5 = 14$ $= (3)*(2)+(1) *(3)+(5)*(1)=6+3+5=14$

The modulus of $\to a$ $veca$ is

$= ∣ ∣ ∣ ∣ \to a ∣ ∣ ∣ ∣ = | | < 2, 3, 1 > | | = \sqrt{{(2)}^{2} + {(3)}^{2} + {(1)}^{2}} = \sqrt{14}$ $=||veca||=||<2,3,1>|| =sqrt((2)^2+(3)^2+(1)^2)=sqrt14$

Therefore,

$p r o j_{\to a} \to b = \frac{14}{14} < 2, 3, 1 >$ $proj_(veca)vecb=14/14<2, 3,1>$

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What is the projection of $< 3, 1, 5 >$ $<3,1,5>$ onto $< 2, 3, 1 >$ $<2,3,1>$ ?

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Explanation:

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