Multiply vectors by scalars

Scalar multiplication means multiplying every component of a vector by the same real number, so the vector’s magnitude is multiplied by the scalar’s absolute value. Geometrically, a positive scalar preserves direction, a negative scalar reverses direction, and zero produces the zero vector; coordinate and directed-segment representations make these relationships explicit and support vector combinations and equations. This treatment is limited to real scalars and ordinary coordinate vectors, not abstract vector spaces or other scalar systems.

Detailed Explanation: Multiply vectors by scalars

To multiply a vector by a scalar, multiply every component of the vector by the same real number.

\mathbf{v}=\begin{pmatrix}a\\b\end{pmatrix}

and the scalar is $k$ , then

k\mathbf{v} = k\begin{pmatrix}a\\b\end{pmatrix} = \begin{pmatrix}ka\\kb\end{pmatrix}.

Worked example

Find $-2\mathbf{v}$ if

\mathbf{v}=\begin{pmatrix}3\\-4\end{pmatrix}.

Step 1: Multiply the first component by $-2$ .

-2(3)=-6

Step 2: Multiply the second component by $-2$ .

-2(-4)=8

Step 3: Write the new vector.

-2\mathbf{v} = -2\begin{pmatrix}3\\-4\end{pmatrix} = \begin{pmatrix}-6\\8\end{pmatrix}.

So,

\boxed{-2\mathbf{v}=\begin{pmatrix}-6\\8\end{pmatrix}}.

The scalar $-2$ has two effects:

The magnitude is multiplied by $\vert -2 \vert =2$ , so the vector is twice as long.
Because the scalar is negative, the direction is reversed.

A positive scalar keeps the direction the same, while multiplying by $0$ gives the zero vector:

0\mathbf{v}=\begin{pmatrix}0\\0\end{pmatrix}.

Learn by doing: Multiply vectors by scalars

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Vectors - Scalar Multiplication - Vector and Multiple to Final Vector

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