Multiplication of a Vector by a Scalar

Multiplication of a Vector by a Scalar

• A scalar is simply a single numerical value, with no direction.
• A vector is a quantity that has both magnitude (size) and direction.

Basics of Scalar Multiplication

• Scalar multiplication involves multiplying the vector by the scalar.
• If a vector v is multiplied by the scalar of k, then the result is another vector, kv, with the following properties:
• If k > 0, the vector kv is in the same direction as v.
• If k < 0, the vector kv is in the opposite direction as v.

Magnitude of the Resultant Vector

• The magnitude (or length) of kv is **

  k

  ** times the magnitude of v.

• This means that the size of the vector changes, but the ratio of the components remains the same.

Properties of Scalar Multiplication

• Scalar multiplication adheres to two important rules, known as the distributive and associative properties:
• The distributive property states that for any vectors v and w, and any scalar k, we have k(v + w)= kv + kw.
• The associative property states that for any vector v and any scalars a and b, we have (ab)v = a(bv).

Important Reminders on Scalar Multiplication

• When a vector is multiplied by a scalar, its size changes by that factor, and its direction may flip, but it otherwise remains parallel to the original vector.
• Any vector, when multiplied by 0, becomes the zero vector, a vector of zero length.
• Multiplying a vector by 1 leaves it unchanged.
• Negative scalars reflect the vector about the origin in addition to scaling them.
• The rules for scalar multiplication mean that you can scale and add vectors in any order without changing the final result.