The direction cosines are obtained by dividing the direction ratios by the square root of the sum of their squares. Here the magnitude = sqrt(1^2 + 2^2 + 2^2) = sqrt(1 + 4 + 4) = sqrt(9) = 3. So the direction cosines are l = 1/3, m = 2/3 and n = 2/3. We can check l^2 + m^2 + n^2 = 1/9 + 4/9 + 4/9 = 9/9 = 1.

l = a/sqrt(a^2+b^2+c^2), etc.

Vector form: the equation of a line passing through a point with position vector a and parallel to a vector b is r = a + lambda b, where lambda is a scalar parameter and r is the position vector of any point on the line. Cartesian form: if the line passes through the point (x1, y1, z1) and has direction ratios a, b, c, then its equation is (x - x1)/a = (y - y1)/b = (z - z1)/c.

r = a + lambda b

If two lines have direction ratios (a1, b1, c1) and (a2, b2, c2), the angle theta between them is given by cos theta = |a1 a2 + b1 b2 + c1 c2| / (sqrt(a1^2 + b1^2 + c1^2) x sqrt(a2^2 + b2^2 + c2^2)). The lines are perpendicular if a1 a2 + b1 b2 + c1 c2 = 0, and they are parallel if a1/a2 = b1/b2 = c1/c2 (their direction ratios are proportional).

cos theta = |a1a2 + b1b2 + c1c2|/(|d1||d2|)

The general Cartesian equation of a plane is Ax + By + Cz + D = 0, where A, B, C are the direction ratios of the normal to the plane and are not all zero. The equation of a plane that makes intercepts a, b and c on the x-, y- and z-axes respectively is x/a + y/b + z/c = 1 (the intercept form). A plane is uniquely determined by three non-collinear points.

Ax + By + Cz + D = 0

The distance of a point (x1, y1, z1) from the plane Ax + By + Cz + D = 0 is |A x1 + B y1 + C z1 + D|/sqrt(A^2 + B^2 + C^2). Here (x1, y1, z1) = (0, 0, 0), A = 3, B = 4, C = 12, D = -26. Distance = |3(0) + 4(0) + 12(0) - 26|/sqrt(3^2 + 4^2 + 12^2) = |-26|/sqrt(9 + 16 + 144) = 26/sqrt(169) = 26/13 = 2 units.

d = |A x1 + B y1 + C z1 + D|/sqrt(A^2 + B^2 + C^2)

The angle between two planes is equal to the angle between their normals. If the planes are A1 x + B1 y + C1 z + D1 = 0 and A2 x + B2 y + C2 z + D2 = 0, then cos theta = |A1 A2 + B1 B2 + C1 C2| / (sqrt(A1^2 + B1^2 + C1^2) x sqrt(A2^2 + B2^2 + C2^2)). The two planes are perpendicular if A1 A2 + B1 B2 + C1 C2 = 0, and parallel if A1/A2 = B1/B2 = C1/C2.

cos theta = |A1A2 + B1B2 + C1C2|/(|n1||n2|)

For two parallel planes Ax + By + Cz + D1 = 0 and Ax + By + Cz + D2 = 0, the distance between them is |D1 - D2|/sqrt(A^2 + B^2 + C^2). Here A = 2, B = -2, C = 1, D1 = 3, D2 = 9. Distance = |3 - 9|/sqrt(2^2 + (-2)^2 + 1^2) = 6/sqrt(4 + 4 + 1) = 6/sqrt(9) = 6/3 = 2 units.

d = |D1 - D2|/sqrt(A^2 + B^2 + C^2)

A linear programming problem (LPP) is a problem of maximising or minimising a linear function of variables subject to certain linear conditions. The objective function is the linear function Z = ax + by that is to be optimised (maximised or minimised). The constraints are the linear inequalities (or equations) in the variables that restrict the values the variables can take; together with the non-negativity restrictions (x >= 0, y >= 0) they define the region of possible solutions.

Z = ax + by (objective function)

The feasible region is the common region determined by all the constraints (including the non-negativity restrictions) of a linear programming problem; it is the set of all points that satisfy every constraint. A feasible solution is any point that lies in the feasible region. An optimal solution is a feasible solution that gives the maximum or minimum value of the objective function, whichever is required.

The corner point method is based on the fact that the optimal value of the objective function (if it exists) occurs at a corner point (vertex) of the feasible region. The steps are: (1) find the feasible region determined by the constraints and identify its corner points; (2) evaluate the objective function Z = ax + by at each corner point; (3) the largest of these values is the maximum and the smallest is the minimum. For an unbounded region, the optimal value must be checked for existence.

A feasible region is said to be bounded if it can be enclosed within a circle (or a rectangle) of finite radius, that is, it does not extend to infinity. It is unbounded if it extends indefinitely (to infinity) in some direction and cannot be enclosed. For a bounded feasible region, the objective function always has both a maximum and a minimum value at the corner points. For an unbounded region, a maximum or a minimum may not exist and must be checked separately.

The feasible region is a triangle bounded by x + y = 4 and the coordinate axes. Its corner points are (0, 0), (4, 0) and (0, 4). Evaluate Z = 3x + 4y at each corner: at (0, 0), Z = 0; at (4, 0), Z = 12; at (0, 4), Z = 16. The maximum value of Z is 16, which occurs at the corner point (0, 4).

Conditional probability is the probability of occurrence of an event A given that another event B has already occurred; it is written as P(A | B). Its formula is P(A | B) = P(A intersection B)/P(B), provided P(B) is not zero. Here P(A intersection B) is the probability that both A and B occur, and P(B) is the probability of B.

P(A | B) = P(A intersection B)/P(B)

Using the conditional probability formula: P(A | B) = P(A intersection B)/P(B) = 0.3/0.6 = 0.5. Similarly, P(B | A) = P(A intersection B)/P(A) = 0.3/0.5 = 0.6. So P(A | B) = 0.5 and P(B | A) = 0.6.

P(A | B) = P(A intersection B)/P(B)

Two events A and B are said to be independent if the occurrence of one does not affect the probability of the occurrence of the other, that is, P(A | B) = P(A) and P(B | A) = P(B). For independent events, the multiplication rule takes the simple form P(A intersection B) = P(A) x P(B). This is used as the test for independence: if P(A intersection B) = P(A) P(B), the events are independent.

P(A intersection B) = P(A) x P(B)

Since A and B are independent, P(A intersection B) = P(A) x P(B) = 0.3 x 0.4 = 0.12. Using the addition theorem, P(A union B) = P(A) + P(B) - P(A intersection B) = 0.3 + 0.4 - 0.12 = 0.58. So P(A intersection B) = 0.12 and P(A union B) = 0.58.

P(A union B) = P(A) + P(B) - P(A intersection B)

Let E1 and E2 be the events of choosing Bag I and Bag II, so P(E1) = P(E2) = 1/2. Let A be the event of drawing a red ball. Then P(A | E1) = 3/7 and P(A | E2) = 5/11. By Bayes' theorem, P(E1 | A) = [P(E1) P(A | E1)] / [P(E1) P(A | E1) + P(E2) P(A | E2)] = [(1/2)(3/7)] / [(1/2)(3/7) + (1/2)(5/11)] = (3/14) / (3/14 + 5/22). Converting to a common denominator (154): 3/14 = 33/154 and 5/22 = 35/154, so the sum = 68/154. Hence P(E1 | A) = (33/154)/(68/154) = 33/68.

P(E1 | A) = P(E1)P(A|E1)/(sum of P(Ei)P(A|Ei))

A random variable is a real-valued function whose value is determined by the outcome of a random experiment; it assigns a number to each outcome of the sample space (for example, the number of heads when two coins are tossed). A probability distribution of a random variable is a table (or description) that lists all the possible values of the random variable together with their corresponding probabilities; the sum of all these probabilities is always 1.

sum of all p(x) = 1

The mean (expectation) of a random variable is E(X) = sum of x times p(x). Here E(X) = 0 x (1/4) + 1 x (1/2) + 2 x (1/4) = 0 + 1/2 + 1/2 = 1. First we may check that the probabilities add to 1: 1/4 + 1/2 + 1/4 = 1. So the mean of X is 1.

E(X) = sum of x p(x)

This is a binomial experiment with n = 3 trials, probability of success (head) p = 1/2 and failure q = 1/2. The binomial probability of r successes is P(X = r) = nCr p^r q^(n - r). For exactly 2 heads (r = 2): P(X = 2) = 3C2 (1/2)^2 (1/2)^1 = 3 x (1/4)(1/2) = 3 x 1/8 = 3/8. So the probability of getting exactly 2 heads is 3/8.

P(X = r) = nCr p^r q^(n-r)