What is a vector?
A scalar is a quantity with magnitude only, such as mass or temperature. A vector has both magnitude and direction, such as displacement or velocity. A vector from point A to point B is written AB (with an arrow above) or with a single letter such as a. In print we may write a in bold or as a with a tilde beneath.
The magnitude (length) of a vector a is written |a|. Two vectors are equal when they have the same magnitude and the same direction, no matter where they are drawn. The negative vector −a has the same magnitude as a but points in the opposite direction, so BA = −AB. The zero vector 0 has magnitude 0 and no defined direction.
Scalar multiplication and parallel vectors
Multiplying a vector a by a scalar k gives ka. The magnitude becomes |k| × |a|. If k > 0 the direction is unchanged; if k < 0 the direction is reversed. Two non-zero vectors are parallel if and only if one is a scalar multiple of the other: a is parallel to b means a = kb for some scalar k.
Key formula
Magnitude of a scalar multiple: |k a| = |k| × |a|. Parallel condition: a = k b for a non-zero scalar k. Negative vector: BA = −AB.
Worked example
A vector a has magnitude |a| = 6. Find |3a| and |−2a|. Using |ka| = |k| × |a|: |3a| = 3 × 6 = 18, and |−2a| = |−2| × 6 = 2 × 6 = 12. The vector −2a points opposite to a.
Remember
- Scalar = magnitude only; vector = magnitude + direction.
- Equal vectors need same magnitude AND same direction.
- Parallel vectors satisfy a = kb.