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What does this mean? Looks like everything is equal since there's nothing in the problem that indicates parallel, perpendicular or $0$?

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    Well, in (b) the answer's obviously $\;\vec b=0\;$ , right? You also have there something related to right triangles, say... – 2017-02-20
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    As a hint for the 3rd question, it tells you that $a^2+b^2=c^2$. Does this look familiar to you? When adding two vectors what shape do they form in conjunction with their sum? – 2017-02-20
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    I don't see anything in problem b that indicates 0. Where do you get it from? The pythag theory? How does that apply to the problem? – 2017-02-20
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    @user416503 vectors can be manipulated algebraically just like other variables or numbers can. If you subtract $\vec{a}$ from both sides of the equation given in part (b), what does that leave you with? If you add $\vec{b}$ to both sides and then divide by two? – 2017-02-20
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    I don't understand. If I subtract a from problem (a), there's nothing left. – 2017-02-20
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    I know about vectors, but I don't understand the way it's presented here. I have a learning disorder. – 2017-02-20
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    Is the assumption that $a$ is the non-negative real number norm $\vec a$? In that case $\vec a + \vec b = \vec c$ while $a + b = c$ is a rare event. ($\vec {(0,1) }+ \vec {(1,0)} = \vec {(1,1)}$ but $1+1 \ne \sqrt 2$) Asking what is implied by them being equal seems like a very clear question. – 2017-02-20
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    Then stop thinking about vectors and start thinking about variables. If you have $x+y=x-y$ does this imply anything about the value of $x$? Does this imply anything about the value of $y$? Can you see why the exact same steps taken to conclude $y$ must be zero implies the same thing about $\vec{b}$ in the second part of this problem? – 2017-02-20
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    So, in a - everything is equal. b - it equals -2b and 0a I don't understand c. – 2017-02-20
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    I assume that "$a+b=c$" means "$|a|+|b|=|c|$" (although I would hardly call this notation standard). Does this help? – 2017-02-20
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    I just came from a tutoring session. The tutor didn't know how to explain it. That's why I'm here. – 2017-02-20
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    Hopefully the first answer, and comments now posted, will help clarify things for you, user416503. – 2017-02-20

2 Answers 2

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  1. the length of vector sum equals to the sum of vector lengths - this means that two vectors parallel

  2. whether you add or subtract vector b, the result does not change - this means that b = 0

  3. the length of vector sum equals "pythagorean sum" of vector lengths - this means that vectors perpendicular.

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    For (1), it's there also a scalar? How's it determined there's parallel? – 2017-02-20
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    I don't understadnd B - if I rearrange, it gives 2b and 0a – 2017-02-20
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    Is 3 just the formula of the problem? – 2017-02-20
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    Second comment above: If you rearrange, you get $2b= 0 \rightarrow b=0$. – 2017-02-20
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    Alex When I commented (Immediately above this comment), I hope you know that I **did not** mean your second response IN your answer; I was referring to the OP's second comment below your answer. – 2017-02-20
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    Length of the sum $c$ of two vectors $a$ and $b$ can be easily derived from the law of cosines: $c^2 = a^2 + b^2 + 2ab \cos \theta$, where $\theta$ is an angle between the two vectors. now, if $c = a + b$, it means $\theta$ must be $0$, if $c^2 = a^2+b^2$, $\theta$ must be 90 deg. – 2017-02-20
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1) Triangle inequality says: $|\vec a + \vec b | \le |\vec a| + |\vec b| $. When does equality hold?

2) $\vec a + \vec b = \vec a - \vec b \implies$

$\vec a - \vec a + \vec b = \vec a - \vec a - \vec b \implies$

$\vec b = -\vec b$.

What does that imply?

3) What does $\vec a + \vec b = \vec c$ mean. Well, the naive and intuitive idea is that if you place $\vec b$ and the endpoint of $\vec a$ and view the vector resulting from the origin of $\vec a$ to the endpoint of $\vec b$ you get a third vector, $\vec c$. $\vec a$, $\vec b$ and $\vec c$ form a triangle with sides of lengths $|a|, |b|$ and $|c|$. (Thats why it is called the triangle inequality.)

Keeping that in mind what does $\vec a +\vec b = \vec c$ so that $|a|^2 + |b|^2 = |c|^2$ imply about the triangle formed? What does that say about the vectors?