Question

# It is given: ∥m∥=4,∥n∥=sqrt{2},⟨m,n⟩=135 find the norm of the vector m+3n.

Vectors and spaces

It is given: $$\displaystyle∥{m}∥={4},∥{n}∥=\sqrt{{{2}}},⟨{m},{n}⟩={135}$$ find the norm of the vector $$m+3n$$.

$$\displaystyle∥{m}+{3}{n}∥^{{{2}}}=⟨{m}+{3}{n},{m}+{3}{n}⟩=⟨{m},{m}⟩+⟨{m},{3}{n}⟩+⟨{3}{n},{m}⟩+⟨{3}{n},{3}{n}⟩=∥{m}∥^{{{2}}}+{6}⟨{m},{n}⟩+{9}∥{n}∥^{{{2}}}={16}+{6}×{135}+{18}={16}+{6}×{135}+{18}={844}$$
$$\displaystyle∥{m}+{3}{n}∥=\sqrt{{{844}}}$$