Addition of Vectors | Vectors jee mains previous year questions

Q 1 :
The angle between vector $\vec{Q}$ and the resultant of $(2 \vec{Q} + 2 \vec{P})$ and $(2 \vec{Q} - 2 \vec{P})$ and is [2024]

$\tan^{- 1} (\frac{2 Q}{P})$
$\tan^{- 1} (\frac{P}{Q})$
$0^{o}$
$\tan^{- 1} (\frac{2 \vec{Q} - 2 \vec{P}}{2 \vec{Q} + 2 \vec{P}})$

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(C) Resultant $\vec{R} = (2 \vec{Q} + 2 \vec{P}) + (2 \vec{Q} - 2 \vec{P})$

$\vec{R} = 4 \vec{Q}$

Angle between $\vec{Q}$ and $\vec{R}$ is zero.

Q 2 :
If two vectors $\vec{A}$ and $\vec{B}$ having equal magnitude $R$ are inclined at an angle $θ$ , then [2024]

$| \vec{A} - \vec{B} | = \sqrt{2} R \sin (\frac{θ}{2})$
$| \vec{A} + \vec{B} | = 2 R \sin (\frac{θ}{2})$
$| \vec{A} + \vec{B} | = 2 R \cos (\frac{θ}{2})$
$| \vec{A} - \vec{B} | = 2 R \cos (\frac{θ}{2})$

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(C) The magnitude of resultant vector

$R^{'} = \sqrt{a^{2} + b^{2} + 2 a b \cos θ}$

Here $a = b = R$

Then $R^{'} = \sqrt{R^{2} + R^{2} + 2 R^{2} \cos θ}$

$= R \sqrt{2} \sqrt{1 + \cos θ} = \sqrt{2} R \sqrt{2 \cos^{2} \frac{θ}{2}} = 2 R \cos \frac{θ}{2}$

Q 3 :
Two forces ${\vec{F}}_{1}$ and ${\vec{F}}_{2}$ are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between ${\vec{F}}_{1}$ and ${\vec{F}}_{2}$ is $\cos^{- 1} (\frac{1}{n}) .$ The value of $| n |$ is ___________ . [2024]

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(6) $let F_{1} = F_{0} then, F_{2} = 3 F_{0} also, F_{R} = 3 F_{0}$

$F_{R}^{2} = F_{1}^{2} + F_{2}^{2} + 2 F_{1} F_{2} \cos θ$

$∴ 3 F_{0} = \sqrt{F_{0}^{2} + 9 F_{0}^{2} + 6 F_{0}^{2} \cos θ}$

$3 F_{0} = F_{0} \sqrt{10 + 6 \cos θ}$

$9 = 10 + 6 \cos θ$

$\cos θ = - \frac{1}{6} \Rightarrow θ = \cos^{- 1} (\frac{1}{- 6})$

$n = - 6 \Rightarrow | n | = 6$

Q 4 :
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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(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 $n = 3$

Topic Question Set

Q 1 :
The angle between vector $\vec{Q}$ and the resultant of $(2 \vec{Q} + 2 \vec{P})$ and $(2 \vec{Q} - 2 \vec{P})$ and is [2024]

Q 2 :
If two vectors $\vec{A}$ and $\vec{B}$ having equal magnitude $R$ are inclined at an angle $θ$ , then [2024]

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Q 4 :
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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Topic Question Set

Q 1 : The angle between vector Q→ and the resultant of (2Q→+2P→) and (2Q→-2P→) and is [2024]

Q 2 : If two vectors A→ and B→ having equal magnitude R are inclined at an angle θ, then [2024]

Q 3 : Two forces F→1 and F→2 are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between F→1 and F→2 is cos-1(1n). The value of |n| is ___________ . [2024]

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Q 4 : 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]

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Q 1 :
The angle between vector $\vec{Q}$ and the resultant of $(2 \vec{Q} + 2 \vec{P})$ and $(2 \vec{Q} - 2 \vec{P})$ and is [2024]

Q 2 :
If two vectors $\vec{A}$ and $\vec{B}$ having equal magnitude $R$ are inclined at an angle $θ$ , then [2024]

Q 4 :
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]