If ${a}$ and ${b}$ are vectors such that $\|a\| = 4$, $\|b\| = 5$, and $\|a + b\| = 7$, then find $\|2a - 3b\|$. If ${a}$ and ${b}$ are vectors such that $\|a\| = 4$, $\|b\| = 5$, and $\|a + b\| = 7$, then find $\|2a - 3b\|$.    
sidenote I'm wondering if $\|a - b\|^2$ is a thing because I know $\| a + b \|^2$ is a thing.
I'm a little confused on how to start. Like, am I suppose magically make $\|a\|$ into $2a$?? Or try to start a path with the number 4, 5, and 7??
Help with be greatly appreciated!!!
 A: $$49=25+16+2\vec{a}\vec{b},$$
which gives $\vec{a}\vec{b}=4$.
Thus,
$$|2\vec{a}-3\vec{b}|=\sqrt{4\cdot16+9\cdot25-12\cdot4}=\sqrt{241}.$$
A: Since $\|a\|=4,\|b\|=5,\|a+b\|=7$, one has
$$ \|a\|^2+2a\cdot b+\|b\|^2=49 $$
from which one obtains
$$ a\cdot b=4. $$
So
$$ \|2a-3b\|^2=4\|a\|^2-12a\cdot b+9\|b\|^2=241 $$
and hence
$$ \|2a-3b\|=\sqrt{241}.$$
A: $$  a \cdot a = 16  $$
$$  b \cdot b = 25 $$
$$ (a + b) \cdot (a + b) = 49 $$
$$ a \cdot a + 2 a \cdot b + b \cdot b = 49 $$
$$  16 + 2 a \cdot b + 25 = 49   $$
$$   2 a \cdot b  = 8  $$
$$    a \cdot b  = 4  $$
$$  (2a-3b) \cdot (2a-3b) = 4 a \cdot a - 12 a \cdot b + 9 b \cdot b  $$
$$  | 2a - 3 b| =  \sqrt {4 a \cdot a - 12 a \cdot b + 9 b \cdot b }  $$
A: Hint: using "$\,\cdot\,$" for the dot product: $$7^2=\|{a} + {b}\|^2 = (a+b)\cdot(a+b) = \|a\|^2+ 2\,a\cdot b + \|b\|^2 = 4^2+5^2+2\,a \cdot b$$
The above gives $\,a \cdot b\,$, which can then be used to determine $\|2a-3b\|^2=(2a-3b)\cdot(2a-3b)$.
A: For this, $||x||^2$ is equivalent to $(x \cdot x)$.  
$$\begin{align}
(7^2) & = ||a + b||^2 \\
 & = (a + b)\cdot(a+b) \\
 & = a \cdot a + b \cdot a + a \cdot b + b \cdot b\\
 & = a \cdot a + 2 (a \cdot b) + b \cdot b \\
 & = ||a||^2 + 2 (a \cdot b) + ||b||^2\\
(7^2) & = (4^2) + 2 (a \cdot b) + (5^2)\\
49 & = 16 + 2 (a \cdot b) + 25
\end{align}$$
So $a \cdot b = 4$
$$\begin{align}
||2a - 3b||^2 & = (2a - 3b) \cdot (2a - 3b)\\
& = (2a \cdot 2a) + (-3b \cdot 2a) + (2a \cdot (-3b)) + ((-3b) \cdot (-3b))\\
& = 4(a \cdot a) - 12 (a \cdot b) + 9 (b \cdot b)\\
& = 4 ||a||^2 - 12 (a \cdot b) + 9 ||b||^2\\
& = 4 (4^2) - 12 (4) + 9(5^2)\\
& = 64 - 48 + 225\\
& = 241
\end{align}$$
So $||2a - 3b|| = \sqrt{241}$
