Two vectors A⃗ and B⃗ are such that A⃗+B⃗=C⃗ and A^2+B^2=C^2. If θis the angle between positive

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Two vectors A⃗ and B⃗ are such that A⃗+B⃗=C⃗ and...

Two vectors $\vec{A}$ and $\vec{B}$ are such that $\vec{A}+\vec{B}=\vec{C}$ and $A^{2}+B^{2}=C^{2}$. If $\theta$ is the angle between positive directions of $\vec{A}$ and $\vec{B}$ then mark the correct alternative (a) $\theta=0^{\circ}$ (b) $\theta=\frac{\pi}{2}$ (c) $\theta=\frac{2 \pi}{3}$ (d) $\theta=\pi$

Two vectors $\vec{A}$ and $\vec{B}$ are such that $\vec{A}+\vec{B}=\vec{C}$ and $A^{2}+B^{2}=C^{2}$. If $\theta$ is the angle between positive directions of $\vec{A}$ and $\vec{B}$ then mark the correct alternative
(a) $\theta=0^{\circ}$
(b) $\theta=\frac{\pi}{2}$
(c) $\theta=\frac{2 \pi}{3}$
(d) $\theta=\pi$

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Key Concepts

Vector Addition

Vector addition involves combining two or more vectors, taking both their magnitudes and directions into account. The resultant vector is obtained by summing the corresponding components of the individual vectors, and its magnitude depends on both the sizes of the original vectors and the angle between them.

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It is calculated as the product of the magnitudes of the two vectors and the cosine of the angle between them. The dot product is essential for determining the angle between vectors, as well as for checking orthogonality.

Perpendicular Vectors

Perpendicular vectors, or orthogonal vectors, are vectors that intersect at a right angle (90°). In terms of the dot product, two vectors are perpendicular if and only if their dot product is zero. This concept is also akin to the Pythagorean theorem, where the square of the magnitude of the hypotenuse equals the sum of the squares of the other two sides when the angle between them is 90°.

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