Scalar Multiplication of Vectors

A vector has both a magnitude (length) and a direction, while a scalar is just a single real number. Scalar multiplication is the operation of multiplying a vector by a scalar to produce a new vector. Every component of the original vector gets multiplied by that same number.

If a vector is written in component form as \(\vec{v} = (a, b)\), then multiplying it by a scalar \(k\) gives:

\(k\vec{v} = k(a, b) = (ka, kb)\)

The result, \(k\vec{v}\), is called a scalar multiple of \(\vec{v}\). This is different from other vector operations like vector addition (combining two vectors) or the dot product (multiplying two vectors together) — scalar multiplication only ever involves one vector and one number.

For a two-dimensional vector \(\vec{v} = (v_1, v_2)\) and scalar \(k\):

\(k\vec{v} = (kv_1, kv_2)\)

For a three-dimensional vector \(\vec{v} = (v_1, v_2, v_3)\), the rule extends naturally:

\(k\vec{v} = (kv_1, kv_2, kv_3)\)

No matter how many components a vector has, scalar multiplication always works the same way: multiply every single entry by \(k\).

Scalar multiplication changes the length of a vector and, depending on the sign of the scalar, may also reverse its direction.

• If \(k > 1\), the vector stretches (gets longer) but keeps the same direction.
• If \(0 < k < 1\), the vector shrinks (gets shorter) but keeps the same direction.
• If \(k = -1\), the vector keeps the same length but points in exactly the opposite direction.
• If \(k < 0\) in general, the vector both changes length and reverses direction.
• If \(k = 0\), the result is the zero vector, which has no direction at all.

The diagram below shows a vector \(\vec{v}\), its scalar multiple \(2\vec{v}\) (twice as long, same direction), and \(-\vec{v}\) (same length, opposite direction).

Multiplying a vector by a scalar \(k\) scales its magnitude by \(|k|\) (the absolute value of the scalar, since length can never be negative):

\(|k\vec{v}| = |k|\,|\vec{v}|\)

For example, if \(\vec{v} = (3, -4)\), then \(|\vec{v}| = \sqrt{3^2 + (-4)^2} = \sqrt{25} = 5\). Multiplying by \(k = 2\) gives \(2\vec{v} = (6, -8)\), and its magnitude is \(|2\vec{v}| = \sqrt{6^2 + (-8)^2} = \sqrt{100} = 10\), which is exactly \(2 \times 5\), confirming the rule.

Scalar multiplication follows several algebraic rules that make it easy to combine with vector addition:

• Distributive over vector addition: \(k(\vec{u} + \vec{v}) = k\vec{u} + k\vec{v}\)
• Distributive over scalar addition: \((k_1 + k_2)\vec{v} = k_1\vec{v} + k_2\vec{v}\)
• Associative: \(k_1(k_2\vec{v}) = (k_1 k_2)\vec{v}\)
• Identity: \(1\vec{v} = \vec{v}\)
• Zero scalar: \(0\vec{v} = \vec{0}\)

These properties are what allow expressions with several vectors and scalars, such as those in vector word problems, to be simplified step by step just like ordinary algebra.

Example 1: Find \(3\vec{v}\) if \(\vec{v} = (2, -5)\).

\(3\vec{v} = 3(2, -5) = (3 \times 2,\ 3 \times (-5)) = (6, -15)\)

Example 2: Find \(-4\vec{u}\) if \(\vec{u} = (-1, 3, 2)\).

\(-4\vec{u} = -4(-1, 3, 2) = (4, -12, -8)\)

Example 3: Simplify \(2(\vec{a} + \vec{b})\) if \(\vec{a} = (1, 4)\) and \(\vec{b} = (3, -2)\).

First add: \(\vec{a} + \vec{b} = (4, 2)\). Then multiply by the scalar: \(2(4, 2) = (8, 4)\).

Two vectors that are scalar multiples of each other with a positive scalar point in the same direction and are considered parallel; this connects directly to the idea of equivalent vectors, since scaling never changes the underlying direction angle unless the scalar is negative. Scalar multiplication is also the tool used to build a unit vector: dividing a vector by its own magnitude (multiplying by the scalar \(\frac{1}{|\vec{v}|}\)) produces a vector of length 1 pointing the same way.

• Forgetting to multiply every component by the scalar, not just the first one.
• Mixing up magnitude scaling: the magnitude scales by \(|k|\), not by \(k\) itself.
• Assuming a negative scalar changes the magnitude in a strange way; it only flips the direction and scales the length by the absolute value of the scalar.

Related lessons

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