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Vectors : Planes, Points, Cross Product and Dot Product

Name: Vectors : Planes, Points, Cross Product and Dot Product
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(1) a. Find the (vector) equation of the plane passing through the points (1,2,-2),
(-1,1,-9), (2,-2,-12).

b. Find the (vector) equation of the plane containing (1,2,-1) and perpendicular to
(3,-1,2).

(2) Suppose a, b, c are non zero vectors.

a. Explain why (a x b) x (a x c) is in the direction of a.

b. Show that (a x b) x (a x c ) = ( a . b x c) a.

c. Show that a x b . (b x c) x (c x a) = (a . b x c)²

A vector equation of a plane is found. Dot and Cross products are calculated. The solution is detailed.

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"Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
"excellent work"
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"Thank you"
"Thank you very much for your valuable time and assistance!"