If and makes an angle with each other, then for . The integer value of is

Question

If $\vec{a}$ and $\vec{b}$ makes an angle $\cos^{- 1} (\frac{5}{9})$ with each other, then $| \vec{a} + \vec{b} | = \sqrt{2} | \vec{a} - \vec{b} |$ for $| \vec{a} | = n | \vec{b} |$ . The integer value of $n$ is __________ . [2024]

Smart Achievers · Accepted Answer

(3) $| \vec{a} + \vec{b} | = \sqrt{2} | \vec{a} - \vec{b} |$

$a^{2} + b^{2} + 2 a b \cos θ = 2 (a^{2} + b^{2} - 2 a b \cos θ)$

$n^{2} b^{2} + b^{2} + 2 n b^{2} \times \frac{5}{9} = 2 n^{2} b^{2} + 2 b^{2} - 4 n b^{2} \times \frac{5}{9}$

$n^{2} + 1 + \frac{10 n}{9} = 2 n^{2} + 2 - \frac{20 n}{9}$

$9 n^{2} - 30 n + 9 = 0 \Rightarrow n = 3, \frac{1}{3}$

Integer value $n = 3$

Solution

Q.
If $\vec{a}$ and $\vec{b}$ makes an angle $\cos^{- 1} (\frac{5}{9})$ with each other, then $| \vec{a} + \vec{b} | = \sqrt{2} | \vec{a} - \vec{b} |$ for $| \vec{a} | = n | \vec{b} |$ . The integer value of $n$ is __________ . [2024]

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Solution

Q. If a→ and b→ makes an angle cos-1(59) with each other, then |a→+b→|=2|a→-b→| for |a→|=n|b→|. The integer value of n is __________ . [2024]

Ans. (3) |a→+b→|=2|a→-b→| a2+b2+2abcosθ=2(a2+b2-2abcosθ) n2b2+b2+2nb2×59=2n2b2+2b2-4nb2×59 n2+1+10n9=2n2+2-20n9 9n2-30n+9=0⇒n=3,13 Integer value n=3

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Q.
If $\vec{a}$ and $\vec{b}$ makes an angle $\cos^{- 1} (\frac{5}{9})$ with each other, then $| \vec{a} + \vec{b} | = \sqrt{2} | \vec{a} - \vec{b} |$ for $| \vec{a} | = n | \vec{b} |$ . The integer value of $n$ is __________ . [2024]