Find the vector components of u 2 0 1 along a 1 2 3 and th

Find the vector components of u = (2, 0, 1) along a = (1, 2, 3) and the vector component of u orthogonal to a. Verify the two vector components you calculated are orthogonal to each other. Use the to determine if f_1(x) = 1, f_2(x) = sin x, and f (x0 = sin 2x are linearly independent.

Solution

In order to show that the two components that we have found are orthogonal or not we need to check if these
two vector components dot product is 0 or not.
If its 0 then it\'ll imply that the two vectors are perpendicular to each other or they are orthogonal to each other

the component of u along a is = <5/14 , 5/7 , 15/14>
the component of u orthogonal to a is = <23/14 , -5/7 , -1/14>

now find the dot product
<5/14 , 5/7 , 15/14> . <23/14 , -5/7 , -1/14> = 115/14^2 - 25/7^2 - 15/14^2 = 5/49*[23/4 - 5 - 3/4]
                                                                            = 5/49*[(23 - 20 - 3)/4]
                                                                            = 5/49*[0] = 0

Since the dot product is 0 so we can say that the two components are orthogonal to each other.