Respuesta :
Answer:
Step-by-step explanation:
Given:
u = 1, 0, -4
In unit vector notation,
u = i + 0j - 4k
Now, to get all unit vectors that are orthogonal to vector u, remember that two vectors are orthogonal if their dot product is zero.
If v = v₁ i + v₂ j + v₃ k is one of those vectors that are orthogonal to u, then
u. v = 0 [substitute for the values of u and v]
=> (i + 0j - 4k) . (v₁ i + v₂ j + v₃ k) = 0 [simplify]
=> v₁ + 0 - 4v₃ = 0
=> v₁ = 4v₃
Plug in the value of v₁ = 4v₃ into vector v as follows
v = 4v₃ i + v₂ j + v₃ k -------------(i)
Equation (i) is the generalized form of all vectors that will be orthogonal to vector u
Now,
Get the generalized unit vector by dividing the equation (i) by the magnitude of the generalized vector form. i.e
[tex]\frac{v}{|v|}[/tex]
Where;
|v| = [tex]\sqrt{(4v_3)^2 + (v_2)^2 + (v_3)^2}[/tex]
|v| = [tex]\sqrt{17(v_3)^2 + (v_2)^2}[/tex]
[tex]\frac{v}{|v|}[/tex] = [tex]\frac{4v_3i + v_2j + v_3k}{\sqrt{17(v_3)^2 + (v_2)^2}}[/tex]
This is the general form of all unit vectors that are orthogonal to vector u
where v₂ and v₃ are non-zero arbitrary real numbers.
[tex]\left ( \frac{1}{\sqrt{17}},0, \frac{-4}{\sqrt{17}}\right )[/tex]
A vector is a quantity that has both magnitude and direction.
Two vectors are said to be orthogonal if they are perpendicular to each other.
Unit vectors orthogonal to [tex]u=\left ( a,b,c \right )[/tex] are given by [tex]\left ( \frac{a}{\sqrt{a^2+b^2+c^2}},\frac{b}{\sqrt{a^2+b^2+c^2}},\frac{c}{\sqrt{a^2+b^2+c^2}} \right )[/tex]
So, vectors orthogonal to [tex]u=\left ( 1,0,-4\right )[/tex] are [tex]\left ( \frac{1}{\sqrt{1^2+0^2+(-4)^2}},\frac{0}{\sqrt{1^2+0^2+(-4)^2}},\frac{-4}{\sqrt{1^2+0^2+(-4)^2}} \right )[/tex]
[tex]\left ( \frac{1}{\sqrt{1^2+0^2+(-4)^2}},\frac{0}{\sqrt{1^2+0^2+(-4)^2}},\frac{-4}{\sqrt{1^2+0^2+(-4)^2}} \right )=\left ( \frac{1}{\sqrt{17}},0, \frac{-4}{\sqrt{17}}\right )[/tex]
So, unit vector orthogonal to [tex]u[/tex] is [tex]\left ( \frac{1}{\sqrt{17}},0, \frac{-4}{\sqrt{17}}\right )[/tex]
For more information:
https://brainly.com/question/17108011?referrer=searchResults
