The sum of the magnitudes of two vectors acti | vedclass

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Question

(A) Let the two vectors be $\vec{A}$ and $\vec{B}$,where $|\vec{A}|$ is the smaller magnitude.Given: $|\vec{A}| + |\vec{B}| = 18$ and $|\vec{R}| = 12$

Vedclass · Accepted Answer

$5, 13$

Vedclass · Answer

(A) Let the two vectors be $\vec{A}$ and $\vec{B}$,where $|\vec{A}|$ is the smaller magnitude.Given: $|\vec{A}| + |\vec{B}| = 18$ and $|\vec{R}| = 12$.Let $|\vec{A}| = x$,then $|\vec{B}| = 18 - x$.Since the resultant $\vec{R}$ is at $90^{\circ}$ to the smaller vector $\vec{A}$,we have the relation: $|\vec{R}|^2 + |\vec{A}|^2 = |\vec{B}|^2$.Substituting the values: $12^2 + x^2 = (18 - x)^2$.$144 + x^2 = 324 + x^2 - 36x$.$36x = 324 - 144$.$36x = 180$.$x = 5$.Thus,$|\vec{A}| = 5$ and $|\vec{B}| = 18 - 5 = 13$.

The sum of the magnitudes of two vectors acting at a point is $18$ and the magnitude of their resultant is $12$. If the resultant is at $90^{\circ}$ with the vector of smaller magnitude,then the magnitudes of the vectors are

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