Two Mutually Perpendicular Vectors
$\Begingroup$ I Cannot Demonstrate This Excersice About Vectors: If $A$ and $B$ Are Mutually Perpendicular Vectors( Different to Zero Vector) and "C" Is Any...
I cannot demonstrate this excersice about vectors: If $A$ and $B$ are mutually perpendicular vectors( different to zero vector) and "c" is any number show that :
$\|A +cB \| \geq \|A\|$.
Well I don´t know where to start, I was thinking about using the dot product but I don´t get how to use it, could anyone give me a solution or any idea?
2 Answers
You need only note that \begin{align*} (A+cB, A+cB) &= (A, A) + c^2(B, B) + 2c^2(A, B)\\ &= (A, A) + c^2(B, B)\\ &\ge (A, A). \end{align*}
Hint
$$\left\|A+cB\right\|^2=\left\|A\right\|^2+|c|^2\left\|B\right\|^2$$
Elena Rostova holds a Master's degree in Public Health Journalism. She covers groundbreaking medical research, holistic wellness trends, mental health awareness, and nutritional science.
