Six vectors \(\vec{a},\vec{b},\vec{c},\vec{d},\vec{e}\) and \(\vec{f}\) have the magnitudes and directions indicated in the figure. Which of the following statements is true?
\(\overline{d}+\overline{e}=\overline{f}\)
Vectors are added by the triangle law: placing the tail of the second vector at the head of the first, the resultant runs from the tail of the first to the head of the second, so its direction is fixed by the directions of the two components.
Step 1 — From the figure, \(\vec{d}\) points vertically upward, \(\vec{e}\) points horizontally to the left, \(\vec{b}\) points downward, \(\vec{c}\) points to the right, and \(\vec{f}\) points diagonally upward and to the left.
Step 2 — Adding an upward vector and a leftward vector gives a resultant pointing up and to the left, matching \(\vec{f}\), so \(\vec{d}+\vec{e}=\vec{f}\).
Step 3 — \(\vec{b}+\vec{e}\) points down and to the left, \(\vec{b}+\vec{c}\) points down and to the right, and \(\vec{d}+\vec{c}\) points up and to the right; none of these has the direction of \(\vec{f}\).
\(\overline{d}+\overline{e}=\overline{f}\) option is correct.
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