A designer plans to create a new pattern of stitching a triangular piece of fabric shown in the figure above. She will stitch fa
bric in a straight line from the vertices of the triangle to the circumcenter of the triangle, located at point C, and then stitch an additional line of fabric back from the circumcenter to each vertex. How much fabric, in inches, will she need in total to create the design? (Note: Answer should be a whole number.)
1 answer:
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Answer:
A), B) and D) are true
Step-by-step explanation:
A) We can prove it as follows:
B) When you compute the product Ax, the i-th component is the matrix of the i-th column of A with x, denote this by Ai x. Then, we have that . Now, the colums of A are orthonormal so we have that (Ai x)^2=x_i^2. Then
.
C) Consider . This set is orthogonal because
, but S is not orthonormal because the norm of (0,2) is 2≠1.
D) Let A be an orthogonal matrix in . Then the columns of A form an orthonormal set. We have that
. To see this, note than the component
of the product
is the dot product of the i-th row of
and the jth row of
. But the i-th row of
is equal to the i-th column of
. If i≠j, this product is equal to 0 (orthogonality) and if i=j this product is equal to 1 (the columns are unit vectors), then
E) Consider S={e_1,0}. S is orthogonal but is not linearly independent, because 0∈S.
In fact, every orthogonal set in R^n without zero vectors is linearly independent. Take a orthogonal set and suppose that there are coefficients a_i such that
. For any i, take the dot product with u_i in both sides of the equation. All product are zero except u_i·u_i=||u_i||. Then
then
.
<h2>Answer:</h2>
We will use substitution to solve this system for x.
Now that we know what x is, we can determine y by pluging in -3 for x in one of the original equations. I will use the first one.
We now have our solution.
D.
We can check our answer by pluging the solution into one of the original equations to see if both sides equal each other. I will use the first equation.
Since the equation checks out, our solution is correct.