Solve for x if the vectors (2, x, 7-x) and (x, 3, -2) are orthogonal
if orthogonal, then their dot product is zero
(2,x,7-x) . (x , 3, -2) = 0
2x + 3x - 14 + 2x = 0
7x = 14
x = 2
check:
(2,2,5).(2,3,-2) = 4 + 6 - 10 = 0
To solve for x, we need to find when the dot product of the two vectors is zero, as two vectors are orthogonal if and only if their dot product is zero.
The dot product of two vectors (a, b, c) and (d, e, f) is given by the formula:
dot product = a*d + b*e + c*f
In our case, the dot product of the vectors (2, x, 7-x) and (x, 3, -2) must be zero:
2*x + x*3 + (7-x)*(-2) = 0
Now we can solve this equation for x:
2x + 3x + (-14 + 2x) = 0
5x - 14 + 2x = 0
7x - 14 = 0
7x = 14
x = 14/7
x = 2
Therefore, the value of x is 2.