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 x, we can use the formula for the 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 have:

||proj_u(v)|| = 6

The norm of the projected vector can be calculated as:

||proj_u(v)|| = sqrt((proj_u(v) dot proj_u(v)))

Therefore, we can set up the equation:

6 = sqrt(((proj_u(v) dot proj_u(v)))

Now let's calculate the dot product of proj_u(v) with itself:

proj_u(v) = ((v dot u) / (u dot u)) * u

v dot u = (x)(4) + (3)(0) + (2)(-3) = 4x - 6

u dot u = (4)(4) + (0)(0) + (-3)(-3) = 16 + 9 = 25

proj_u(v) dot proj_u(v) = (((v dot u) / (u dot u)) * u) dot (((v dot u) / (u dot u)) * u)
= ((4x - 6) / 25) * (4, 0, -3) dot ((4x - 6) / 25) * (4, 0, -3)
= ((4x - 6) / 25) * (16x^2 - 48x + 36)

Now we can substitute this into our equation:

6 = sqrt(((4x - 6) / 25) * (16x^2 - 48x + 36))

We can square both sides to get rid of the square root:

(6)^2 = (((4x - 6) / 25) * (16x^2 - 48x + 36))

36 = (4x - 6) * (16x^2 - 48x + 36) / 25

Multiplying both sides by 25 to eliminate the denominator:

25 * 36 = (4x - 6) * (16x^2 - 48x + 36)

900 = (4x - 6) * (16x^2 - 48x + 36)

Expanding the right side:

900 = 64x^3 - 192x^2 + 144x - 96x^2 + 288x - 216

Simplifying:

900 = 64x^3 - 288x^2 + 432x - 216

Rearranging the equation to set it equal to zero:

64x^3 - 288x^2 + 432x - 1116 = 0

Now we can solve this cubic equation to find the value of x.