What is the projection of $< - 1, 5, 7 >$ $< -1, 5, 7>$ onto $< 7, - 3, - 5 >$ $< 7,-3, -5>$ ?

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Answer 1 · 2016-12-21T14:41:30

The answer is $= - \frac{57}{83} ⟨ 7, - 3, - 5 ⟩$ $=-57/83〈7,-3,-5〉$

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

$= \frac{\to a . \to b}{∥ ∥ \to a {∥ ∥}^{2}} \to a$ $=(veca.vecb)/(∥veca∥^2)veca$

The dot product is $\to a . \to b = ⟨ - 1, 5, 7 ⟩ . ⟨ 7, - 3, - 5 ⟩$ $veca.vecb=〈-1,5,7〉.〈7,-3,-5〉$

$= (- 7 - 15 - 35) = - 57$ $=(-7-15-35)=-57$

The modulus of $\to a$ $veca$ is $= ∥ \to a ∥ = ∥ ⟨ 7, - 3, - 5 ⟩ ∥ ∥$ $=∥veca∥=∥〈7,-3,-5〉∥$

$= \sqrt{49 + 9 + 25} = \sqrt{83}$ $=sqrt(49+9+25)=sqrt83$

The vector projection is $= - \frac{57}{83} ⟨ 7, - 3, - 5 ⟩$ $=-57/83〈7,-3,-5〉$

What is the projection of < -1, 5, 7><−1,5,7>< -1, 5, 7> onto < 7,-3, -5><7,−3,−5>< 7,-3, -5>?