For what values of c is ||c(1,2,3)|| = 1?

Question

I just wanted to check if what I'm doing for a particular problem is correct. My background: business student with average math skills taking a linear algebra class. Would appreciate it if someone confirms what I'm doing and/or points me to the right direction!

Question: For what values of $c$ is $||c(1,2,3)|| = 1?$

Here's what I have so far:

$$||(c,2c,3c)|| = 1$$ $$\sqrt{(c)^{2}+(2c)^{2}+(3c)^{2}} = 1$$ $$\sqrt{c^{2}+4c^{2}+9c^{2}} = 1$$ $$\sqrt{14c^{2}} = 1$$ $$\sqrt{14}\cdot c = 1$$ $$c = \sqrt{\frac{1}{14}}$$

Thanks a lot!

Answer 1

Be careful $$\forall x \in \mathbf{R}, \sqrt{x^2} = |x|$$ In general, $|x| \neq x$.

Here you have $$ |c| = \sqrt{\frac{1}{14}} $$ so $$ c = \sqrt{\frac{1}{14}} \text{ or } c = -\sqrt{\frac{1}{14}} $$