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September 21st, 2017, 11:53 AM   #1
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Problems about linear combination, direction numbers, and transposes

Hello everyone,

I have some math homework here I can't quite understand:

Q3: (a) Express the given vector as a linear combination of base vectors:
(b) Find the direction numbers and direction cosines of each vector.
c) Find the transpose of each vector.

i) v = [2]
[5].
ii) v = [2]
[-2]
[3]

How would I go about solving these 3 questions? A simple step solution is enough. Don't need to go into too much detail.

Thanks!
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September 21st, 2017, 01:22 PM   #2
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Aside from (b), I can say these questions are not difficult. What have you tried? Do you need to find out definitions of the terms used in the questions?

As for (b), I'm not familiar with "direction numbers"
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September 21st, 2017, 01:22 PM   #3
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You need to clarify the question. (a,b,c) all refer to vectors (given, base, each). What are the vectors?
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September 21st, 2017, 03:44 PM   #4
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I mainly need help on subquestion b.
The vectors are [2, 5] for i and [2, -2, 3] for ii.

So find the direction vectors and direction cosines for vector [2, 5] as well as [2, -2, 3]
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September 21st, 2017, 05:15 PM   #5
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This page will help. It explains a method for calculating direction cosines in three-space as well as some general information. I suggest you take a look, if you're willing.
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September 22nd, 2017, 04:06 AM   #6
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Problem (a) says "Express the given vector as a linear combination of base (basis) vectors". What basis vectors? Any vector space has an infinite number of bases. Do you mean the standard basis vectors for $\displaystyle R^2$ and $\displaystyle R^3$?

"Direction cosines" of a vector are the cosines of the angles the vector makes with the coordinate axes. They are also the components of a unit vector in the same direction. The "direction numbers" are the direction cosines multiplied a factor to give the correct length to the vector. They are also the components of the vector.
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