Given that P-Q-R- 0- Two out of the three vectors are equal in magnitude- The magnitude of the third vector is -2 time that of the other two- Which of the following can be the angles these vectors make with each other-

The correct option is A 90^∘, 135^∘, 135^∘ R⃗ = ( P⃗ + Q⃗ #160;) Let, P⃗ = p _ 1 î + p _ 2 ĵ Q⃗ = q _ 2 î + q _ ...

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Standard XII

Mathematics

Direction Cosines

Given that ...

Question

Given that $\to P + \to Q + \to R = 0.$ Two out of the three vectors are equal in magnitude. The magnitude of the third vector is $\sqrt{2}$ time that of the other two. Which of the following can be the angles these vectors make with each other?

90^{\circ}, 135^{\circ}, 135^{\circ}

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45^{\circ}, 45^{\circ}, 90^{\circ}

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30^{\circ}, 60^{\circ}, 90^{\circ}

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45^{\circ}, 90^{\circ}, 135^{\circ}

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Solution

The correct option is A $90^{\circ}, 135^{\circ}, 135^{\circ}$
$\to R = - (\to P + \to Q) L e t, \to P = p_{1}^i + p_{2}^j \to Q = q_{2}^i + q_{2}^j \to R = r_{1}^i + r_{2}^j \sqrt{{r_{1}}^{2} + {r_{2}}^{2}} = \sqrt{2 ({p_{1}}^{2} + {p_{2}}^{2})} = \sqrt{2 ({q_{1}}^{2} + {q_{2}}^{2})} r_{1} = - (p_{1} + q_{1}) r_{2} = - (p_{2} + q_{2}) \Rightarrow \sqrt{{r_{1}}^{2} + {r_{2}}^{2}} = \sqrt{{p_{1}}^{2} + {q_{1}}^{2} + {p_{2}}^{2} + {q_{2}}^{2} + 2 p_{1} q_{1} + 2 p_{2} q_{2}} \Rightarrow \sqrt{2 ({p_{1}}^{2} + {p_{2}}^{2})} = \sqrt{2 ({p_{1}}^{2} + {p_{2}}^{2}) + 2 p_{1} q_{1} + 2 p_{2} q_{2}} \Rightarrow 2 p_{1} q_{1} + 2 p_{2} q_{2} = 0 \Rightarrow \to P \cdot \to Q = 0 \Rightarrow ∠ \to P \to Q = \frac{π}{2} ∠ \to R \to P = θ_{1} ∠ \to R \to Q = θ_{2} cos θ_{1} = \frac{\to R \cdot \to P}{\sqrt{2} P} = \frac{- (\to P + \to Q) \cdot \to P}{\sqrt{2} P} = - \frac{1}{\sqrt{2}} \Rightarrow θ_{1} = \frac{3 π}{4} cos θ_{2} = \frac{\to R \cdot \to Q}{\sqrt{2} Q} = \frac{- (\to P + \to Q) \cdot \to Q}{\sqrt{2} Q} = - \frac{1}{\sqrt{2}} \Rightarrow θ_{2} = \frac{3 π}{4}$

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Given that →P+→Q+→R=0. Two out of the three vectors are equal in magnitude. The magnitude of the third vector is √2 time that of the other two. Which of the following can be the angles these vectors make with each other?

Given that $\to P + \to Q + \to R = 0.$ Two out of the three vectors are equal in magnitude. The magnitude of the third vector is $\sqrt{2}$ time that of the other two. Which of the following can be the angles these vectors make with each other?