What is the projection of $< 3, - 7, 4 >$ $<3,-7,4>$ onto $< 1, 4, 0 >$ $<1,4,0 >$ ?

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Answer 1 · 2016-11-22T07:37:58

The projection is $⟨ - \frac{25}{17}, - \frac{100}{17}, 0 ⟩$ $〈-25/17,-100/17,0〉$

Let $\to b = ⟨ 3, - 7, 4 ⟩$ $vecb=〈3,-7,4〉$ and $\to a = ⟨ 1, 4, 0 ⟩$ $veca=〈1,4,0〉$

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 $\to a . \to b$ $veca.vecb$

$= ⟨ 3, - 7, 4 ⟩ . ⟨ 1, 4, 0 ⟩ = 3 \cdot 1 - 7 \cdot 4 + 0 = - 25$ $=〈3,-7,4〉.〈1,4,0〉=3*1-7*4+0=-25$

The modulus of $\to a = ∥ 1, 4, 0 ∥ = \sqrt{1 + 16} = \sqrt{17}$ $veca=∥1,4,0∥=sqrt(1+16)=sqrt17$

So the vector proection is

$= - \frac{25}{17} ⟨ 1, 4, 0 ⟩ = ⟨ - \frac{25}{17}, - \frac{100}{17}, 0 ⟩$ $=-25/17〈1,4,0〉=〈-25/17,-100/17,0〉$

What is the projection of <3,−7,4><3,-7,4> onto <1,4,0><1,4,0 >?