8. Let α, β, γ be the angles made by a vector u ∈ R 3 with positive x, y, z−axis respectively. Then, the numbers l = cos α, m =
cos β, n = cos γ are called the direction cosines of u.
(a) Find the formulas for the direction cosines: l, m, n of the vector u = hu1, u2, u3i.
(b) Prove that l 2 + m2 + n 2 = 1.
(c) Find the direction cosines l, m, n of the vector u = h1, −1, 1i.
(a) Find the formulas for the direction cosines: l, m, n of the vector u = hu1, u2, u3i.
(b) Prove that l 2 + m2 + n 2 = 1.
(c) Find the direction cosines l, m, n of the vector u = h1, −1, 1i.
1 answer:
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One way of solving this problem can be Heron's formula. It is a formula that allows us to compute the area of a triangle, knowing the length of its three sides.
For the sake of clarity, let's assume that the point in the top-left lies in the origin, and let's call it point A. Then, the "middle" point is point B, and the bottom-right point is point C.
If we fix the coordinate axis with the origin in A, we have the following coordinates for the three points:
We can compute the length of any side using the formula for the distance between two points:
Plugging the approriate values, we get the following lenghts:
Now that we have the lengths, we can use Heron's formula: given the side lenghts and the semiperimeter
, the area is given by
If you plug our values, you will get an area of 6.5. So, unless I'm mistaken, none of the answers seem to mach, whereas 7 seems the best approximation.
Step-by-step explanation:
7. There is no tree diagram, sticking it as impossible to solve at the moment.
8. The temperature went down 5 degrees and then goes up 7 degrees. The low temperature is -5.
9. The formula is , or 1/8