If-vector-a-b-c-0-a-7-b-3-and-c-5-find-the-angle-vector-a-and-c-

Question Number 117543 by bemath last updated on 12/Oct/20

If vector \vec{a} + \vec{b} + \vec{c} = 0

∣ \vec{a} ∣= 7, ∣ \vec{b} ∣= 3 and ∣ \vec{c} ∣= 5

find the angle vector \vec{a} and \vec{c} ?

Answered by bobhans last updated on 12/Oct/20

let α = ∡ (\vec{a}, \vec{c})

by Cosine of law

\Rightarrow \cos (π - α) = \frac{7^{2} + 5^{2} - 3^{2}}{2.7.5}

\Rightarrow - \cos α = \frac{65}{70}; \cos α = - \frac{13}{14}

\Rightarrow α = \cos^{- 1} (- \frac{13}{14}) = 158.21 °

Answered by AbduraufKodiriy last updated on 12/Oct/20

\vec{a} + \vec{c} = - \vec{b} \Rightarrow ∣ \vec{a} + \vec{c} ∣=∣ \vec{b} ∣ \Rightarrow

\Rightarrow \sqrt{∣ \vec{a} ∣^{2} + 2 \cdot \vec{a} \cdot \vec{c} + ∣ \vec{c} ∣^{2}} = 3

\sqrt{7^{2} + 2 \cdot ∣ \vec{a} ∣∣ \vec{b} ∣ c o s (φ) + 5^{2}} = 3

74 + 2 \cdot 7 \cdot 5 c o s (φ) = 9 \Rightarrow 70 c o s (φ) = - 65 \Rightarrow

\Rightarrow c o s (φ) = - \frac{13}{14} \Rightarrow φ = π - a r c c o s (\frac{13}{14})

φ i s a n g l e v e c t o r \vec{a} a n d \vec{c}