Question

A projeção ortogonal do vetor (3, 5, -4) sobre o vetor (2, -4, -1) é um vetor (x, y, z). Qual o valor de z? ---resposta com 4 casas decimais.

Answer 1

Resposta:

$\text{proj}^{(3,5,-4)}_{(2,-4,-1)}=\left(-\frac{20}{21},\frac{40}{21},\frac{10}{21}\right)\cong (-0,9524;1,9048;0,4762)$

Explicação passo a passo:

Temos que:

$\text{proj}^{(3,5,-4)}_{(2,-4,-1)}=\frac{(3,5,-4)\cdot(2,-4,-1)}{(2,-4,-1)\cdot(2,-4,-1)}(2,-4,-1)$

$\text{proj}^{(3,5,-4)}_{(2,-4,-1)}=\frac{3\cdot2+5\cdot(-4)+(-4)\cdot(-1)}{2\cdot2+(-4)\cdot(-4)+(-1)\cdot(-1)}(2,-4,-1)$

$\text{proj}^{(3,5,-4)}_{(2,-4,-1)}=\frac{6-20+4}{4+16+1}(2,-4,-1)$

$\text{proj}^{(3,5,-4)}_{(2,-4,-1)}=-\frac{10}{21}(2,-4,-1)$

$\text{proj}^{(3,5,-4)}_{(2,-4,-1)}=\left(-\frac{20}{21},\frac{40}{21},\frac{10}{21}\right)\cong (-0,9524;1,9048;0,4762)$