In two dimensional case the linear optimization (linear programming) is specified as follows: Find the values $(x, y)$ such that the goal function $$g(x, y) = a x + b y \;\;\; (Eq. 1)$$ is maximized (...

Let xc, yc be the coordinates of the center of the rectangle. Translate your points such that the center is the new origin: xt = x1 - xc; yt = y1 - yc; Rotate around the origin by the angle a: c = ...

The notation := was used in some early programming languages. The meaning of x := 3 was "assign 3 to the variable x". So, the assignment could be distinguished from the the test if (x = 3) .... In ...

I am refering to the picture below, where the rotation angle is denoted by $\alpha$. We have for the new $x', y'$ if $0 \le \alpha \le 90$:: $$x' = BD = BC + CD = FE \cos(90-\alpha) + CD = (h - y) \... View answer 7 votes Let us denote by "old" the usual cartesian system with orthogonal axes and by "new" the system with the skew axes (\alpha_1, \alpha_2)^T, (\beta_1, \beta_2)^T (expressed in the old system). An old ... View answer 7 votes This is indeed an exercise in semantics as mentioned by @Asaf. My interpretation would be:$$fine(\text{me}) \land flue(\text{him})$$where fine, flue are predicates and me, him are constants. ... View answer Accepted answer 6 votes For 2 vectors a = (a_x, a_y), b = (b_x, b_y) compute$$c_z = a_x b_y - a_y b_x$$If c_z > 0 then b is on the CCW side of a. If c_z < 0 then b is on the CW side of a. If c_z = 0 ... View answer Accepted answer 6 votes The fundamental problem here is to find two vectors u, v that are orthogonal to the vector n = (n_x, n_y, n_z)^T. Morevover, the procedure should be numerically stable. I assume that |n| = 1. ... View answer 5 votes Let us denote by X_n the number of possible words of length n ending with the letter X, and similarly Y_n, Z_n. Then the reccurence applies:$$X_{n+1} = X_n + Y_n + Z_nY_{n+1} = X_n + Z_n$... View answer 5 votes Suppose that the vectors$v_1, ..., v_n$are$m$-dimensional. Build an$m \times n$matrix$A$such that the vectors$v_i$are columns of$A$. The problem is now formulated as follows. Find the$n$-... View answer Accepted answer 5 votes $$wh(d) = \begin{cases} h \cos(d) + w \sin(d), & \mbox{if} \;\; 0^\circ \le d \le 90^\circ \;\; \mbox{or} \;\; 180^\circ \le d \le 270^\circ \;\; \\ w \cos(d-90) + h \sin(d-90), & \mbox{... View answer Accepted answer 4 votes The linear program can be formulated just using the binary 0/1-variables x_{rcn} with the meaning x_{rcn} = 1 if and only if there is the number n in the cell rc, i.e in the row r and the ... View answer 4 votes I do not have any explicit formula (this could be hard), but invested some time into an enumeration program. This is a nice exercise in backtracking. The heart of the algorithm is the recursive ... View answer 4 votes The problem has already been analyzed by @joriki. You can get a numerical solution by trying an online Simplex solver. Just google for it. I tried the first one that I found: Simplex Method Tool and ... View answer Accepted answer 4 votes Here is an attempt based on my experiences with furniture moving. The long dimension a=4.3 will surely be horizontal. One of the short dimensions, call it b will be vertical, the remaining dimension c ... View answer Accepted answer 3 votes Your considerations about the quality of the intersection of two planes are correct. The intersection will be computed most accurately if the planes are orthogonal. The interection is not defined for ... View answer Accepted answer 3 votes This is a slightly different question to this one. But the answer is the same: it is not possible to determine a or b or a+b. Look at this figure: We can move the red lines anywhere in the \... View answer 3 votes I will use the notation of @Day and consider only the right half of the hull (x \ge 0). We have$$\cos \alpha = k, \;\;\; \sin \alpha = \sqrt{1 - k^2}, \;\;\; y_0 = r/k$$So, the equation of the ... View answer 3 votes Let us denote by L = L(a, b, c) the line ax + by + c = 0. (a, b, c) can be considered a 3-dimensional vector v and all lines can be represented by the 3-dimensinal vector space R^3. The ... View answer 3 votes The task is to prove \neg s from the assumptions (1)-(4). Proof 1 (by deduction, Modus Ponens): These are true facts: \neg p by (4) o by (1) \neg r by (3) in the equivalent form \neg o \... View answer Accepted answer 3 votes Let us first consider the two dimensional case to explain the concepts. A line segment of the length L (I will call it line L) is to be laid down on a unit square grid such that the number of the ... View answer 3 votes A more symmetric factorization would be:$$A^2 + B^2 + C^2 - 2AB -2AC - 2BC =(A + B + C)^2 - 4AB - 4AC - 4BC =(A + B + C - 2 \sqrt{AB + AC + BC}) (A + B + C + 2 \sqrt{AB + AC + BC})$$if$$... View answer 3 votes Your problem can be reduced to the well known problem of picking a uniformly distributed point on a unit sphere. There are various methods for solving this problem, see e.g. www.cgafaq.info or ... View answer 3 votes You will find simple formulae in http://en.wikipedia.org/wiki/Polygon. Look for the various formulae A = ... in the section Area and centroid. You will find the appropriate formula depending whether ... View answer 2 votes Notice that Fred can always invest in 1-year deposits and get this money at the beginning of the next year for further investments. This means that all available money will always be invested and we ... View answer Accepted answer 2 votes The discrepancy is caused by taking the maximal derivative in the interval [2.5, 3.0]: $$k = \max |g'(x)| = |g'(2.5)| = \sqrt{10}/5 = 0.632$$ So, you assume that the solution error is ... View answer Accepted answer 2 votes If you impose$m$linear and linearly independent constraints on$\mathbb R^n$, then the set has the dimension$n-m$. The vector space of these$m$constraints has the dimension$m$. If only$k, k \...

To obtain the instantaneous velocity, compute the derivative $f'(t) = d f(t) / dt$. The velocity at $t=5$ is $f'(5)$.

Besides using the Newton's identities as mentioned in @Soarer's comment, you could also consider an algorithm to generate all combinations. E.g. to compute $$abc+abd+acd+bcd$$ you would generate 3-...
You can compute $x$ and $h$. (BTW, your second equation should be $\cos(\alpha) = (D+x)/a$.). Even then, you cannot determine $\alpha$. You just got the point $S$. According to the Intercept theorem ...