Knowing u = (4,0,-3), v = (x,3,2) and that the orthogonal projection of v on u is a vector of norm 6, determine x.
Thank you
To determine the value of x, we can use the formula for the orthogonal projection of vector v onto vector u:
proj_u(v) = ((v dot u) / (u dot u)) * u
Given that the orthogonal projection of v onto u has a norm of 6, we can write the equation:
||proj_u(v)|| = ||((v dot u) / (u dot u)) * u|| = 6
Now, let's substitute the given values into the equation:
||((v dot u) / (u dot u)) * u|| = 6
First, let's calculate the dot product of v and u:
v dot u = (x)(4) + (3)(0) + (2)(-3)
= 4x - 6
Next, let's calculate the dot product of u and u:
u dot u = (4)(4) + (0)(0) + (-3)(-3)
= 16 + 9
= 25
Now, we can rewrite the equation as:
||((4x - 6) / 25) * (4, 0, -3)|| = 6
Since the magnitude of a vector is equal to the square root of the sum of its squared components, we can simplify the equation further:
sqrt((4x - 6)^2 + 0^2 + (-3)^2) = 6
Squaring both sides of the equation, we get:
(4x - 6)^2 + 9 = 36
Expanding the equation, we have:
16x^2 - 48x + 36 + 9 = 36
Simplifying and moving all terms to one side, we have:
16x^2 - 48x + 9 = 0
Now, we can solve this quadratic equation for x by factoring, completing the square, or using the quadratic formula.
Using the quadratic formula, we have:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
Plugging in the values for a, b, and c, we get:
x = (-(-48) ± sqrt((-48)^2 - 4(16)(9))) / (2(16))
Simplifying further:
x = (48 ± sqrt(2304 - 576)) / 32
x = (48 ± sqrt(1728)) / 32
x = (48 ± 24sqrt(3)) / 32
Therefore, the possible values for x are:
x = (48 + 24sqrt(3)) / 32
x = 3 + sqrt(3)
or
x = (48 - 24sqrt(3)) / 32
x = 3 - sqrt(3)
So, the possible values for x are 3 + sqrt(3) and 3 - sqrt(3).