[Solved] Calculus-Homework 10

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Problem 1

Prove the following identities for vectors a,b,c ∈ R3.

  1. The “BAC–CAB-identity”

a × (b × c) = b(a · c) − c(a · b). (1)

  1. The Jacobi identity in three dimensions

a × (b × c) + b × (c × a) + c × (a × b) = 0.

Problem 2

Prove the following identities for vectors a,b,c,d ∈ R3.

  1. The Cauchy–Binet formula in three dimensions

(a × b) · (c × d) = (a · c)(b · d) − (a · d)(b · c).

Hint: Use the identity u · (v × w) = v · (w × u).

  1. The identity ka × bk2 = kak2 kbk2 − (a b)2 .

Problem 3

  1. Find the minimum distance between the point p = (2,4,6) and the line

x .

  1. Express the equation for the plane that contains the point p and the line x in parametric form. Then proceed to find the vector normal to this plane.

Bonus

Prove the following statement: Let v1,…,vn be linearly independent. If a vector w can be written

w,

then the choice of the coefficients α1,…,αn is unique.

Hint: Recall that a set of vectors is said to be linearly independent if w = 0 implies that all of the coefficients αk = 0 .

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[Solved] Calculus-Homework 10
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