Scalar Multiplication of a Vector

We've discussed vector addition and subtraction. Now, we'll explore scalar multiplication, an operation that combines a vector and a scalar (a regular number).

Recall that multiplication of scalars can be interpreted as repeated addition. For example, $4 \times 3 = 4 + 4 + 4$ . We use a similar idea to define scalar multiplication of a vector. If we have a vector $\overrightarrow{X}$ , then $3 \times \overrightarrow{X}$ (written as $3\overrightarrow{X}$ ) is defined as:

$3\overrightarrow{X} = \overrightarrow{X} + \overrightarrow{X} + \overrightarrow{X}$

This operation is called scalar multiplication because we are multiplying a vector by a scalar.

Geometric Interpretation

Geometrically, scalar multiplication affects the magnitude, but not the direction, of a vector.

In the above example we multiplied the vector $\overrightarrow{X}$ by $3$ , the result is a vector that was pointing in the same direction but it $3$ times longer. If we changed the scalar from $3$ to $4$ then the result would be $4$ times longer, and so on.

We can generalise this idea to define multiplication for scalars that are not whole numbers (for example $0.5$ or $\pi$ ).

For instance, $0.5\overrightarrow{X}$ is a vector half the length of $\overrightarrow{X}$ but pointing in the same direction, while $\pi\overrightarrow{X}$ is a vector $\pi$ times the length of $\overrightarrow{X}$ , with the direction unchanged.

Scalar Multiplication by a Negative Number or Zero

Multiplying a vector by a negative scalar scales its magnitude by the absolute value of the scalar and reverses its direction. In other words, multiplying by $-k$ (where k is a positive scalar) results in a vector k times the length of $\overrightarrow{X}$ but pointing in the opposite direction.

Multiplying a vector by zero results in a vector with zero magnitude – the zero vector, $\overrightarrow{0}$ .

Interactions between Scalar Multiplication and Vector Addition

Scalar multiplication interacts with vector addition and scalar addition in the following ways (these are important properties):

Distributivity over Vector Addition

$k(\overrightarrow{X} + \overrightarrow{Y}) = k\overrightarrow{X} + k\overrightarrow{Y}$

This means that scaling the sum of two vectors by a scalar is the same as scaling each vector individually and then adding the scaled vectors.

Distributivity over Scalar Addition

$(k + l)\overrightarrow{X} = k\overrightarrow{X} + l\overrightarrow{X}$

This means that scaling a vector by the sum of two scalars is the same as scaling the vector by each scalar individually and then adding the resulting scaled vectors.

Associativity of Scalar Multiplication

$k(l\overrightarrow{X}) = (kl)\overrightarrow{X}$

This means that scaling a vector by one scalar and then scaling the result by another scalar is the same as scaling the original vector by the product of the two scalars.

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