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Precalculus · Question of the Day

Precalculus Question of the Day

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Thursday, September 17, 2026

Vectors v⃗\vec v and w⃗\vec w are drawn tail-to-tail. Which correctly shows the head-to-tail construction for v⃗−w⃗\vec v-\vec w using subtraction as addition of the opposite (i.e., v⃗−w⃗=v⃗+(−w⃗)\vec v-\vec w=\vec v+(-\vec w))?

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Vectors v⃗\vec v and w⃗\vec w are drawn tail-to-tail. Which correctly shows the head-to-tail construction for v⃗−w⃗\vec v-\vec w using subtraction as addition of the opposite (i.e., v⃗−w⃗=v⃗+(−w⃗)\vec v-\vec w=\vec v+(-\vec w))?

  1. Draw v⃗\vec v, then draw w⃗\vec w starting at the head of v⃗\vec v; the resultant is v⃗−w⃗\vec v-\vec w.
  2. Draw v⃗\vec v, then draw −w⃗-\vec w starting at the head of v⃗\vec v; the resultant from the tail of v⃗\vec v to the head of −w⃗-\vec w is v⃗−w⃗\vec v-\vec w. (correct answer)
  3. Draw w⃗\vec w, then draw −v⃗-\vec v starting at the head of w⃗\vec w; the resultant is v⃗−w⃗\vec v-\vec w.
  4. Draw v⃗\vec v and w⃗\vec w tail-to-tail; v⃗−w⃗\vec v-\vec w is the vector from the tip of v⃗\vec v to the tip of w⃗\vec w.

Explanation: This question tests understanding of vector subtraction and how to represent it graphically using the tip-to-tip method. Vector subtraction is defined as adding the opposite: v - w = v + (-w), where -w is the vector with the same magnitude as w but pointing in the opposite direction, having components -w = ⟨-w₁, -w₂⟩. To find v - w graphically, we can rewrite it as v + (-w): first draw v, then draw -w (the opposite of w), place the tail of -w at the head of v, and the resultant from the tail of v to the head of -w is v - w. Choice B is correct because it properly describes the tip-to-tip method. Choice D describes the graphical vector going from the tip of v to the tip of w, but v - w goes from the tip of w to the tip of v (the opposite direction). For the tip-to-tip graphical method, place both vectors tail-to-tail and draw the difference vector from the tip of what you're subtracting (w) to the tip of what you're subtracting from (v). Remember that vector subtraction is not commutative: v - w and w - v are different vectors pointing in opposite directions, so order matters in subtraction unlike addition.