# How to normalize a vector which is combined with other vectors?

if $$\vec v_1$$ and $$\vec v_2$$ are both $$3$$ by $$1$$ matrix ,now i new vector $$f_A$$

$$\vec f_A=\alpha \vec v_1+\beta \vec v_2$$,and if i want to normalize the $$\vec f_A$$ to let $$\vec f_A ^T\vec f_A=1$$,how do i rewrite the formula

$$\vec n_{f_A}=\frac{\vec f_A}{||\vec f_A||}=\frac{\alpha \vec v_1+\beta \vec v_2}{\sqrt{\alpha^2 \vec v_1^T\vec v_1+\beta^2 \vec v_2 ^T\vec v_2}}?$$

Because i found that $$\vec n_{f_A} \vec n_{f_A}^T \neq 1 ,$$so it must be wrong somewhere in my normalization of $$\vec f_A$$

• Why is $\| f_A \| = \sqrt{\alpha^2v_1^tv_1+\beta^2v_2^tv_2}$? Oct 8 '19 at 2:08
• @Azif00 because i want to find the length of $f_A$,that is ,the square of the inner product of $f_A$ Oct 8 '19 at 2:25

It should be $$\hat{\vec {f}}_A=\frac{\vec{f}_A}{\sqrt{(\vec{f}_A)^T \vec{f}_A}},$$ where $$T$$ denotes transpose. In denominator $$v_1^T v_2$$ and $$v_2^T v_1$$ will also be there. In case, these two are orthogonal these terms will vanish.