Fix the mistake: Basic Plane Vectors

Plane Vectors ① What Is a Vector?

The “numbers” we have studied so far represent quantities with magnitude only, but many quantities in the world have both “magnitude” and “direction,” such as velocity, force, and displacement. A tool that represents these quantities together is a vector.

Directed Line Segments and Vectors

A line segment connecting two points A and B with a direction from A to B is called the directed line segment AB. A is its initial point and B is its terminal point. A vector is obtained by removing “position” from a directed line segment and focusing only on its direction and magnitude; it is written as →AB.

  • Directed line segments with the same direction and equal magnitude are treated as the same vector, even when they are placed at different positions
  • The magnitude (length) of a vector is written as |→AB|
  • A vector with magnitude 0 is called the zero vector →0 (its direction is not considered)
  • A vector with the same magnitude as →AB but the opposite direction is called the opposite vector, written as −→AB or →BA
Example ①
In parallelogram ABCD, which vector is equal to →AB?
Solution:In a parallelogram, AB ∥ DC, |AB|=|DC|, and the directions A→B and D→C are the same. Therefore, →DC.

Equal, Parallel, and Unit Vectors

Two vectors →a and →b are equal when they have equal magnitude and the same direction: →a = →b. They are parallel when they have the same or opposite directions: →a ∥ →b. A vector with magnitude 1 is called a unit vector; the unit vector in the same direction as →a is →a/|→a|.

Example ②
Let O be the center of regular hexagon ABCDEF. List all vectors parallel to →OA, excluding →OA itself.
Solution:→OD has the opposite direction to →OA, and →AO and →DO are also parallel. Therefore, →OD, →AO, →DO.
Key Point
  • A vector has “direction” and “magnitude”; its position does not matter
  • Equal: same magnitude and same direction
  • Parallel: same or opposite direction
  • →0 has magnitude 0 and no considered direction; a unit vector has magnitude 1

Practice Problems

  1. In rectangle ABCD, give a vector equal to →AB and a vector opposite in direction to →AB.
  2. Let P, Q, and R be the midpoints of the sides of equilateral triangle ABC, where P is the midpoint of BC, Q is the midpoint of CA, and R is the midpoint of AB. Give a vector equal to →AR.
  3. If |→a|=3, express the unit vector in the same direction as →a using →a.

Solutions and Explanations

  1. Solution:Equal: →DC; opposite direction: →BA, →CD
    Explanation: In a rectangle, AB ∥ DC and their lengths are equal, while A→B and D→C have the same direction.
  2. Solution:→RB
    Explanation: Since R is the midpoint of AB, AR=RB and the directions are the same.
  3. Solution:→a / 3
    Explanation: The unit vector is →a/|→a|. Substitute |→a|=3.

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