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Six vectors, $\vec{a}$ through $\vec{f}$ have the magnitudes and directions indicated in the figure. Which of the following statements is true?
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A
$\vec{b} + \vec{e} = \vec{f}$
B
$\vec{b} + \vec{c} = \vec{f}$
C
$\vec{d} + \vec{c} = \vec{f}$
D
$\vec{d} + \vec{e} = \vec{f}$
Detailed Solution
By the triangle law, if two vectors are drawn head to tail, their sum is the vector drawn from the tail of the first to the head of the second.
Using the magnitudes and directions shown in the figure, place the tail of $\vec{e}$ at the head of $\vec{d}$.
The vector from the tail of $\vec{d}$ to the head of $\vec{e}$ then has the same magnitude and direction as $\vec{f}$.
The other combinations ($\vec{b} + \vec{e}$, $\vec{b} + \vec{c}$, $\vec{d} + \vec{c}$) give resultants that do not coincide with $\vec{f}$ in direction.
Hence $\vec{d} + \vec{e} = \vec{f}$.
Using the magnitudes and directions shown in the figure, place the tail of $\vec{e}$ at the head of $\vec{d}$.
The vector from the tail of $\vec{d}$ to the head of $\vec{e}$ then has the same magnitude and direction as $\vec{f}$.
The other combinations ($\vec{b} + \vec{e}$, $\vec{b} + \vec{c}$, $\vec{d} + \vec{c}$) give resultants that do not coincide with $\vec{f}$ in direction.
Hence $\vec{d} + \vec{e} = \vec{f}$.