Question

Use Theorem 5.5.2 to write the vector 3−5−5 as a linear combination of u,

v, and w. Note that u, v, and w are orthonormal. Use Parseval's formula to compute...

a) u+v+w
b) u−v−w
c) 2u+3v−4w
d) 4u−2v+3w

Answer 1

To write the vector 3-5-5 as a linear combination of u, v, and w, we can use Theorem 5.5.2. The vector can be written as 3u - 5v - 5w. Using Parseval's formula, we can compute different combinations of u, v, and w.

Step-by-step explanation:

To write the vector 3-5-5 as a linear combination of u, v, and w, we can use Theorem 5.5.2. The theorem states that any vector can be expressed as a linear combination of orthonormal vectors. In this case, u, v, and w are orthonormal, meaning they are all perpendicular to each other and have a magnitude of 1.

Let's write 3-5-5 as a linear combination of u, v, and w:

3-5-5 = a*u + b*v + c*w

Since u, v, and w are orthonormal, their dot products are:

u*u = 1, v*v = 1, w*w = 1, u*v = 0, u*w = 0, v*w = 0

Using these dot products, we can solve for a, b, and c:

a = (3-5-5)*u = 3*u = 3

b = (3-5-5)*v = -5*v = -5

c = (3-5-5)*w = -5*w = -5

Therefore, the vector 3-5-5 can be written as a linear combination of u, v, and w as 3u - 5v - 5w.

Now, let's compute a few examples using Parseval's formula:

a) u+v+w = 1*u + 1*v + 1*w = 1u + 1v + 1w = u + v + w

b) u-v-w = 1*u + (-1)*v + (-1)*w = 1u - 1v - 1w = u - v - w

c) 2u+3v-4w = 2*u + 3*v + (-4)*w = 2u + 3v - 4w

d) 4u-2v+3w = 4*u + (-2)*v + 3*w = 4u - 2v + 3w