If two vectors $\vec{A}$ and $\vec{B}$ are at an angle $\theta \neq 0^\circ$, then
[amp_mcq option1=”$|\vec{A}|+|\vec{B}|=|\vec{A}+\vec{B}|$” option2=”$|\vec{A}|+|\vec{B}|>|\vec{A}+\vec{B}|$” option3=”$|\vec{A}|+|\vec{B}|<|\vec{A}+\vec{B}|$” option4=”$|\vec{A}|+|\vec{B}|=|\vec{A}-\vec{B}|$” correct=”option2″]
This question was previously asked in
UPSC CDS-1 – 2019
For any two vectors $\vec{A}$ and $\vec{B}$, the triangle inequality states that the magnitude of their sum is less than or equal to the sum of their magnitudes: $|\vec{A}+\vec{B}| \le |\vec{A}|+|\vec{B}|$. Equality holds only when the vectors are collinear and point in the same direction, which corresponds to the angle between them $\theta = 0^\circ$. If $\theta \neq 0^\circ$, the vectors form two sides of a triangle, and the sum vector forms the third side. In a triangle, the length of any side is strictly less than the sum of the lengths of the other two sides. Therefore, if $\theta \neq 0^\circ$, $|\vec{A}+\vec{B}| < |\vec{A}|+|\vec{B}|$, or equivalently, $|\vec{A}|+|\vec{B}| > |\vec{A}+\vec{B}|$.