find-the-angle-between-3i-4j-and-i-j-

Question Number 39172 by Rio Mike last updated on 03/Jul/18

f i n d t h e a n g l e b e t w e e n

3 i - 4 j a n d i - j

Commented by math khazana by abdo last updated on 03/Jul/18

l e t \vec{u} = 3 i - 4 j \Rightarrow u (3, - 4)

\vec{v} = i - j ? \Rightarrow \vec{v} (1, - 1)

c o s (\vec{u}, \vec{v}) = \frac{\vec{u} . \vec{v}}{∣∣ \vec{u} ∣∣ . ∣∣ \vec{v} ∣∣} = \frac{3 \times 1) + (4)}{5 \sqrt{2}} = \frac{7}{5 \sqrt{2}}

s i n (\vec{u}, \vec{v}) = \frac{d e t (u, v)}{∣∣ u ∣∣ . ∣∣ v ∣∣} = \frac{| \begin{matrix} 3 1 \\ - 4 - 1 \end{matrix} |}{5 \sqrt{2}} = \frac{1}{5 \sqrt{2}} \Rightarrow

t a n (u, v) = \frac{1}{7} \Rightarrow θ = a r c t a n (\frac{1}{7}) .

Answered by tanmay.chaudhury50@gmail.com last updated on 03/Jul/18

v e c t o r 3 i - 4 j m a k e a n g l e α w i t h x a x i s

s o m_{1} = t a n α = \frac{- 4}{3}

m_{2} = t a n β = \frac{- 1}{1}

t a n θ = \frac{m_{1} \sim m_{2}}{1 + m_{1} m_{2}} = \frac{- 1 + \frac{4}{3}}{1 + \frac{4}{3}} = \frac{\frac{1}{3}}{\frac{7}{3}} = \frac{1}{7}

θ = {t a n}^{- 1} (\frac{1}{7})

c o s θ = \frac{7}{\sqrt{50}}

a n o t h e r a p p r o a c h \dots

c o s θ = \frac{3 \times 1 + (- 4) \times (- 1)}{\sqrt{3^{2} + {(- 4)}^{2}} \times \sqrt{1^{2} + {(- 1)}^{2}}}

= \frac{7}{\sqrt{50}}