text: # Cauchy-Schwarz inequality with zero angle?
  
  Cauchy-Schwarz Inequality:
  
  If $$\textbf{u}$$ and $$\textbf{v}$$ are vectors in a real inner product space $$V$$, then $$|\left\langle\textbf{u},\textbf{v}\right\rangle|\leq||\textbf{u}||\ ||\textbf{v}||$$
  
  What will happen with the Cauchy-Schwarz inequality if the angle between the two vectors is zero?
  
  • Algebraically, what does it mean for the angle between $u$ and $v$ to be $0$? Answer this and the answer to your question will present itself. Nov 11, 2014 at 16:37
  • @GitGud $|<\textbf{u},\textbf{v}>| = ||\textbf{u}|| ||\textbf{v}||$ ? I thought this, but it seemed a bit simple, considering the space left open in the past question paper in which one could answer it :) Nov 11, 2014 at 16:39
  
  the definition of the angle $\alpha\in[0,\pi]$ between $u,v$ is: $$\cos\alpha = \frac{\langle u,v\rangle}{\|u\| \|v\|}$$
  so $\alpha =0$ iff $$1=\frac{\langle u,v\rangle}{\|u\| \|v\|}\iff \langle u,v\rangle = \|u\| \|v\|$$
  When angle between $v,w$ is $0$ (in other words $v=\alpha w$ or $w=\alpha v$ for some $\alpha$), then (if $w = \alpha v$):
  $$|\langle w,v\rangle|=|\langle \alpha v,v\rangle|=|\alpha\langle v,v\rangle|=|\alpha|\|v\|^2=|\alpha|\|v\|\|v\|=\|\alpha v\|\|v\|=\|w\|\|v\|$$
  The same with $v=\alpha w$.
explanation: The document discusses the Cauchy-Schwarz inequality and specifically explores what happens when the angle between two vectors is zero. It includes a question about the algebraic meaning of the angle being zero and provides an answer, explaining that the inner product of the vectors equals the product of their norms. The document also includes a detailed derivation to support this conclusion.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Cauchy-Schwarz inequality with zero angle?
  
  Cauchy-Schwarz Inequality:
  
  If $$\textbf{u}$$ and $$\textbf{v}$$ are vectors in a real inner product space $$V$$, then $$|\left\langle\textbf{u},\textbf{v}\right\rangle|\leq||\textbf{u}||\ ||\textbf{v}||$$
  
  What will happen with the Cauchy-Schwarz inequality if the angle between the two vectors is zero?
  
  • Algebraically, what does it mean for the angle between $u$ and $v$ to be $0$? Answer this and the answer to your question will present itself. Nov 11, 2014 at 16:37
  • @GitGud $|<\textbf{u},\textbf{v}>| = ||\textbf{u}|| ||\textbf{v}||$ ? I thought this, but it seemed a bit simple, considering the space left open in the past question paper in which one could answer it :) Nov 11, 2014 at 16:39
  
  the definition of the angle $\alpha\in[0,\pi]$ between $u,v$ is: $$\cos\alpha = \frac{\langle u,v\rangle}{\|u\| \|v\|}$$
  so $\alpha =0$ iff $$1=\frac{\langle u,v\rangle}{\|u\| \|v\|}\iff \langle u,v\rangle = \|u\| \|v\|$$
  When angle between $v,w$ is $0$ (in other words $v=\alpha w$ or $w=\alpha v$ for some $\alpha$), then (if $w = \alpha v$):
  $$|\langle w,v\rangle|=|\langle \alpha v,v\rangle|=|\alpha\langle v,v\rangle|=|\alpha|\|v\|^2=|\alpha|\|v\|\|v\|=\|\alpha v\|\|v\|=\|w\|\|v\|$$
  The same with $v=\alpha w$.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document discusses the Cauchy-Schwarz inequality and specifically explores what happens when the angle between two vectors is zero. It includes a question about the algebraic meaning of the angle being zero and provides an answer, explaining that the inner product of the vectors equals the product of their norms. The document also includes a detailed derivation to support this conclusion.
  
  Label: question-and-answer