questions below:

Determine if the following vectors are orthogonal. 10 2 UE 5 V= 5 Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or a simplified fraction.) O A. The vectors u and v are orthogonal because u . v = O B. The vectors u and v are not orthogonal because u . v = O C. The vectors u and v are not orthogonal because u + v = O D. The vectors u and v are orthogonal because u + v =21: The space is C[0,27l:] and the inner product is (f,g) = If(t)g(t) dt. Show that sin mt and cos nt are orthogonal for all positive integers m and n. 0 Begin by writing the inner product using the given functions. 21E (sin mt, cos nt) = I [D] dt 0 Use a trigonometric identity to write the integrand as a sum of sines. 21: (sin mt, cos nt) = % I [D] dt 0 Then integrate with respect to t. 21: (sin mt, cos nt) = $11] 0 Evaluate the result at the end points of the interval. Note that m - n in the denominator means that this result does not apply to m = n. 1 (sin mt, cos nt) = E [ (D) - (DH (Simplify your answers.) Then simplify this result to get the inner product for all positive integers m a5 n. (sin mt, cos nt) = D Write the inner product for the case m = n and integrate. 21: J- sin mt cos mt dt = [D] I? 0 Evaluate the integral. 2:: I sin mt cos mt dt = D (Simplify your answer.) 0 Therefore, sin ml and cos nt are orthogonal for all positive integers m and n because the inner product is always |:|