Let a → = i + 2 j + 3 k and b → =...

chivistaelmore

chivistaelmore

Answered

2022-07-22

Let a = i + 2 j + 3 k and b = 2 i + 5 k. For which value of t when 2 t 2 holds the length of the vector c t = t a + ( 1 t ) b is as small as possible?
How should one approach this? The minimum would be when c t is perpendicular to some vector d = b a or is there something else Im not seeing?

Answer & Explanation

Jazlene Dickson

Jazlene Dickson

Expert

2022-07-23Added 15 answers

Hint:
Write out the magnitude as a function of t and then differentiate to find the maximum.
Pierre Holmes

Pierre Holmes

Expert

2022-07-24Added 2 answers

Write c t = t ( a b ) + b . Then,
| c t | 2 = | a b | 2 t 2 + 2 ( a b ) b t + | b | 2 = | a b | 2 ( t + ( a b ) b | a b | 2 ) 2 ( ( a b ) b ) 2 | a b | 2 + | b | 2 ( | a b | | b | ) 2 ( ( a b ) b ) 2 | a b | 2 = | c t | m i n 2

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