1.23. Express the unit vector directed toward the point (x, Y, z) from an arbitrary point in the plane y--5.
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
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VECTOR ANALYSIS
11
1.20. Given the plane 4x + 3y + 2z = 12, find the unit vector normal to the surface in the direction away
from the origin.
Ans. (4a, + 3n, + 2n,)/V20
1.21. Find the relationship which the cartesian components of A and B must satisfy if the vector fields are
everywhere parallel.
A
Ans.
В, В, В,
1.22. Express the unit vector directed toward the origin from an arbitrary point on the line described
by x=0, y=3.
-3m, – zm,
V9 + z
Ans.
1.23. Express the unit vector directed toward the point (x,, y, z.) from an arbitrary point in the
plane y --5.
(x, – x)m, + (y, + 5)a, + (z, - z)m,
V(x, – x)' + (y, + 5)' + (z, – z)
Ans
1.24. Expreas the unit vector dirccted toward the point (O,0, h) from an arbitrary point in the plane --2.
-Xn, - ya, + (h + 2)m,
V + y' + (h + 2)?
Ans.
Given A-Sn, and B-4a, + B,,, find B, such that the angle between A and B is 45°. If B also
has a term B,",, what relationship must exist between B, and B,?
1.25.
Ans. B, - ±4, VB; + B; = 4
1.26. Show that the absolute value of A•B×C is the volume of the parallelepiped with edges A, B, and C.
(Hint: First show that the base has area (BXC].)
1.27. Given A-2a, -,, B-3m, +a,, and C=-2a, + 6a, - 4a, show that C is 1 to both A and B.
1.28. Given A-, -a,, B= 2n,, and C=-, + 3a,, find A BXC. Examine other variations of
this scalar triple product.
Ans. -4. t4
1.29. Using the vectors of Problem 1.28 find (AX B) x C.
Ans.
-8a,
1.30. Find the unit vector directed from (2, –5, –2) toward (14, –5, 3).
5
, +
13
12
Ans.
13
1.31. Find the vector directed from (10, 3x/4, x/6) to (5, x/4, x), where the endpoints are given in spherical
coordinates.
Ans.
-9.66m, – 3.54m, + 10.61a,
1.32. Find the distance between (2, x/6, 0) and (1, A, 2), where the points are given in cylindrical
coordinates.
Ans. 3.53
1.33. Find the distance between (1, x/4, 0) and (1, 3x/4, x), where the points are given in spherical
coordinates.
Ans. 2.0
1.34. Use spherical coordinates and integrate to find the area of the region 0spsa on the spherical shell
of radius a. What is the result when a= 2x?
Ans. 2aa', A = 4ta?
1.35. Use cylindrical coordinates to find the area of the curved surface of a right circular cylinder of radius a
and height h.
Ans. 2nah](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F582a9576-fb09-4101-9ada-f607e6cbdf83%2Ff7a51d24-4993-44d6-9524-ed91f96211e5%2F5amy04_processed.jpeg&w=3840&q=75)
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