Question

Find all unit vectors that are orthogonal to the vector u = 1, 0, −4 .

Answer 1

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

$\frac{v}{|v|}$

Where;

|v| = $\sqrt{(4v_3)^2 + (v_2)^2 + (v_3)^2}$

|v| = $\sqrt{17(v_3)^2 + (v_2)^2}$

$\frac{v}{|v|}$ = $\frac{4v_3i + v_2j + v_3k}{\sqrt{17(v_3)^2 + (v_2)^2}}$

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.