Question

if vectors l P l , l Q l and l R l have magnitudo 5, 12, 13 units and l P l+l Q l= l R l, the angle between l Q l and l R l is
a. cos^-1 5/12
b. cos^-1 5/13
c. cos^-1 12/13
d. cos^-1 7/13

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Answer 1

Stated that , The vectors l P l , l Q l & l R l have magnitude 5 , 12 & 13 units , respectively and | P | + | Q | = | R | .

Need To Calculate : The angle between l Q l and l R l ?

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━

❍ Let's, say that θ be angle between l Q l and l R l .

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀¤ Finding ❝ angle between l Q l and l R l ❞ :

Given that ,

$\qquad :\implies \sf | \: P \:|\:+\:|\:Q\: | \: =\:|\:R\:|\: \\\\ \qquad :\implies \sf | \: P \:|\: =\:|\:R\:|\:|\:-\:|\:Q\: | \: \\\\ \qquad :\implies \sf \: P^2 \:\: =\:\:R^2\:+\:Q^2\:\:-\:2 \: Q R \:cos \: \theta \:$

⠀⠀⠀⠀⠀⠀⠀Where ,

• | P | = 5 units ,

• | Q | = 12 units &,

• | R | = 13 units

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}$

$\qquad :\implies \sf \: P^2 \:\: =\:\:R^2\:+\:Q^2\:\:-\:2 \: Q R \:cos \: \theta \: \\\\ \qquad :\implies \sf \: (5)^2 \:\: =\:\:(12)^2\:+\:(13)^2\:\:-\:2 \:(5)(13) \:cos \: \theta \: \\\\ \qquad :\implies \sf \: (5)^2 \:\: =\:\:(12)^2\:+\:(13)^2\:\:-\:312 \:cos \: \theta \: \\\\ \qquad :\implies \sf \: 25 \:\: =\:\:144\:-+\:169\:-\:312\:cos \: \theta \: \\\\ \qquad :\implies \sf \: 25 \:\: =\:\:313\:-\:312\:cos \: \theta \: \\\\\qquad :\implies \sf \: 25 \:-\:313\: =\:\:\:-\:312\:cos \: \theta \: \\\\ \qquad :\implies \sf \: -288 \:\: =\:\:\:-\:312\:cos \: \theta \: \\\\ \qquad :\implies \sf \: 288 \:\: =\:\:\:\:312\:cos \: \theta \: \\\\ \qquad :\implies \sf \: \:cos \: \theta\:=\:\dfrac{288}{312} \: \\\\ \qquad :\implies \sf \: \:cos \: \theta\:=\:\dfrac{12}{13} \: \\\\\qquad :\implies \underline {\boxed{{\frak{\purple { \theta \:=\:cos^{-1} \bigg(\dfrac{12}{13}\bigg) }}}}}\: \: \bigstar \:$

$\qquad \therefore \underline {\sf Hence, \: The \:Angle \:between \:\:|\:Q\: | \: \&\:|\:R\:|\:is\:\pmb{\bf cos^{-1} (12/13)\:}\:.}$