Solve Equation 5x - ||v|| v = ||w||(w-5x)

[Fernando Revilla]

I quote an unsolved question from MHF posted by user Civy71 on February 18th, 2013

Solve the equation 5x - ||v|| v = ||w||(w-5x) for x with v = (1, 2, -4, 0) and w = (-3, 5, 1, 1)

 

[Fernando Revilla]

We have:

$
\begin{aligned}5x-||v||v=||w||(w-5x)&\Leftrightarrow 5x+5||w||x=||w||w+||v||v\\&\Leftrightarrow 5(1+||w||)x=||w||w+||v||v\\&\Leftrightarrow x=\dfrac{1}{5(1+||w||)}(||w||w+||v||v)
\end{aligned}
$

Now, substitute $v=(1, 2, -4, 0)$, $w=(-3, 5, 1, 1 )$, $||w||=6$, $||v||=\sqrt{21}$.

 

[HallsofIvy]

Personally, I would have immediately substituted the given values for v and w.

5x - ||v|| v = ||w||(w-5x) for x with v = (1, 2, -4, 0) and w = (-3, 5, 1, 1)
so $$||v||= \sqrt{1+ 4+ 16}= \sqrt{21}$$ and $$||w||= \sqrt{9+ 25+ 1+ 1}= \sqrt{38}$$

Let x= (w, x, y, z) so the equation becomes
$$(5w, 5x, 5y, 5z)- (\sqrt{21}, 2\sqrt{21}, -4\sqrt{21}, 0)= (-3\sqrt{38}, 5\sqrt{38}, \sqrt{38}, \sqrt{38})- (5\sqrt{38}w, 5\sqrt{38}x, 5\sqrt{38}y, 5\sqrt{38}z)$$
which gives the four numeric equations
$$5w- \sqrt{21}= -3\sqrt{38}- 5\sqrt{38}w$$
$$5x- 2\sqrt{21}= 5\sqrt{38}- 5\sqrt{38}x$$
$$5y+ 4\sqrt{21}= \sqrt{38}- 5\sqrt{38}y$$
$$5z= \sqrt{38}- 5\sqrt{38}z$$

 

[Fernando Revilla]

Personally, I would have immediately substituted the given values for v and w.

All right, no problem with that. My preference was for the sake of generalization.

 

[blue_raver22]

To solve this equation, we will first need to define the notation used. ||v|| represents the magnitude or length of the vector v, which is calculated by taking the square root of the sum of the squares of its components. In this case, ||v|| = √(1^2 + 2^2 + (-4)^2 + 0^2) = √21. Similarly, ||w|| = √(3^2 + 5^2 + 1^2 + 1^2) = √36 = 6.

Substituting these values into the equation, we get:

5x - √21(1, 2, -4, 0) = 6(-3, 5, 1, 1 - 5x)

Expanding the equation, we get:

5x - (√21)(1, 2, -4, 0) = (-18, 30, 6, 6) + (30x, -50x, -10x, -10x)

Combining like terms, we get:

5x - (30x, -50x, -10x, -10x) = (-18, 30, 6, 6) + (√21)(1, 2, -4, 0)

Simplifying, we get:

-25x = (-18 + √21, 30 - 2√21, 6 + 4√21, 6)

Dividing both sides by -25, we get:

x = (18 - √21)/25, (2√21 - 30)/25, (4√21 - 6)/25, -6/25

Therefore, the solution to the equation is x = (18 - √21)/25, (2√21 - 30)/25, (4√21 - 6)/25, or -6/25.