Given that P⃗+Q⃗+R⃗=0. Two out of the three vectors are...

Given that $\vec{P}+\vec{Q}+\vec{R}=0$. Two out of the three vectors are equal in magnitude. The magnitude of the third vector is $\sqrt{2}$ times that of the other two. Which of the following can be the angles between these vectors? (A) $90^{\circ}, 135^{\circ}, 135^{\circ}$ (B) $45^{\circ}, 45^{\circ}, 90^{\circ}$ (C) $30^{\circ}, 60^{\circ}, 90^{\circ}$ (D) $45^{\circ}, 90^{\circ}, 135^{\circ}$

Given that $\vec{P}+\vec{Q}+\vec{R}=0$. Two out of the three vectors are equal in magnitude. The magnitude of the third vector is $\sqrt{2}$ times that of the other two. Which of the following can be the angles between these vectors?
(A) $90^{\circ}, 135^{\circ}, 135^{\circ}$
(B) $45^{\circ}, 45^{\circ}, 90^{\circ}$
(C) $30^{\circ}, 60^{\circ}, 90^{\circ}$
(D) $45^{\circ}, 90^{\circ}, 135^{\circ}$

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Key Concepts

Vector Equilibrium

This concept refers to the condition where a set of vectors adds up to zero. In a closed system (like a triangle), this means that the tail of one vector meets the head of the next, and together they form a closed path. It is essential when establishing relationships between the magnitudes and the directions (angles) of the vectors in equilibrium.

Dot Product and Angle Determination

The dot product of two vectors provides a connection between their magnitudes and the cosine of the angle between them. Through the relation (v · w = |v||w| cos?), one can deduce the angles when the magnitudes are given or vice versa. This is critical in problems involving the direction of vectors and their geometric configuration.

Law of Cosines in Vector Geometry

When three vectors form a closed polygon, the law of cosines can be applied to relate the magnitudes of the vectors and the angles between them. This approach simplifies the analysis of vector relationships in equilibrium, allowing one to derive conditions under which certain angle measures, such as right angles or obtuse angles, occur.

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