Given that, 𝐀+𝐁+𝐂=0 . Out of three vectors, two are equal in...

Given that, $\mathbf{A}+\mathbf{B}+\mathbf{C}=0 .$ Out of three vectors, two are equal in magnitude and the magnitude of third vector is $\sqrt{2}$ times that of either of the two having equal magnitude. Then, the angles between vectors are given by (a) $45^{\circ}, 45^{\circ}, 90^{\circ}$ (b) $90^{\circ}, 135^{\circ}, 135^{\circ}$ (c) $30^{\circ}, 60^{\circ}, 90^{\circ}$ (d) $45^{\circ}, 60^{\circ}, 90^{\circ}$

Given that, $\mathbf{A}+\mathbf{B}+\mathbf{C}=0 .$ Out of three vectors, two are equal in magnitude and the magnitude of third vector is $\sqrt{2}$ times that of either of the two having equal magnitude. Then, the angles between vectors are given by
(a) $45^{\circ}, 45^{\circ}, 90^{\circ}$
(b) $90^{\circ}, 135^{\circ}, 135^{\circ}$
(c) $30^{\circ}, 60^{\circ}, 90^{\circ}$
(d) $45^{\circ}, 60^{\circ}, 90^{\circ}$

Key Concepts

Law of Cosines in Vector Triangles

In a triangle formed by the three vectors in equilibrium, the law of cosines can be applied to relate the side lengths (which correspond to the magnitudes of the vectors) to the angles of the triangle. This provides a method to calculate the angles when the lengths of the sides are known, even if one side is different in magnitude from the other two.

Special Triangle Properties

When two sides of a triangle are equal in length and the third side is ?2 times that length, the triangle has distinctive properties. These properties reveal that one angle between the equal sides is 90° (a right angle), with the other angles determined by the triangle's geometry. Such relationships are useful in deducing the angles between vectors when given specific magnitude ratios.

Vector Equilibrium

When three vectors sum to zero, they are in equilibrium, which means that when placed head-to-tail they form a closed triangle. This condition provides a geometric relationship among the vectors and allows one to use triangle properties to analyze the magnitudes and angles between them.

Dot Product and Angle Calculation

The dot product of two vectors is given by the product of their magnitudes and the cosine of the angle between them. This relationship is key to finding the angles between vectors when their magnitudes and their sums (or differences) are known. In problems where vectors sum to zero, applying the dot product can simplify the determination of the mutual angles.

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