Scalar multiplication of vectors is a fundamental operation in linear algebra and physics where a vector is multiplied by a scalar quantity.
Understanding the Basics
Before diving into the process, let's quickly define the terms:
- Vector: A quantity that has both magnitude (size) and direction. Examples include force, velocity, and displacement. Vectors are often represented graphically as arrows or numerically as ordered lists of numbers (e.g., (x, y) in 2D, (x, y, z) in 3D).
- Scalar: A quantity that only has magnitude. Examples include mass, temperature, time, and speed. Scalars are just numbers.
The Process of Scalar Multiplication
When you perform scalar multiplication, you take a scalar (a number) and multiply it by a vector. The key outcome, as stated in the reference, is that:
"The result of multiplying a vector by a scalar is a vector quantity."
This new vector shares a relationship in direction with the original vector, but its magnitude is changed (scaled) by the scalar.
How it Affects Magnitude and Direction
- Magnitude Change: The magnitude of the resulting vector is the product of the absolute value of the scalar and the magnitude of the original vector. If the scalar is greater than 1, the vector gets longer; if between 0 and 1, it gets shorter; if 0, it becomes the zero vector (zero magnitude, no specific direction).
- Direction Change: The direction depends on the sign of the scalar:
- If the scalar is positive, the resulting vector has the same direction as the original vector.
- If the scalar is negative, the resulting vector has the opposite direction to the original vector.
- If the scalar is zero, the result is the zero vector.
Summary of Effects:
| Scalar (s) | Effect on Magnitude (||v||) | Effect on Direction |
| :--------- | :-------------------------- | :------------------ |
| s > 1 | Becomes s ||v|| (Larger) | Same as original |
| s = 1 | Remains ||v|| | Same as original |
| 0 < s < 1 | Becomes s ||v|| (Smaller) | Same as original |
| s = 0 | Becomes 0 | Result is zero vector |
| s < 0 | Becomes |s| * ||v|| (Larger) | Opposite to original |
As the reference highlights: "The vector has the same direction, but the magnitude changes when multiplied by a scalar." This is true when the scalar is positive. When the scalar is negative, the direction flips, becoming the opposite direction.
Calculation Example
If you have a vector $\vec{v} = \begin{pmatrix} x \ y \end{pmatrix}$ and a scalar $s$, the scalar multiplication is performed by multiplying each component of the vector by the scalar:
$s \vec{v} = s \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} s \cdot x \ s \cdot y \end{pmatrix}$
Numerical Example:
Let vector $\vec{a} = \begin{pmatrix} 2 \ 3 \end{pmatrix}$ and scalar $s = 4$.
$4 \vec{a} = 4 \begin{pmatrix} 2 \ 3 \end{pmatrix} = \begin{pmatrix} 4 \cdot 2 \ 4 \cdot 3 \end{pmatrix} = \begin{pmatrix} 8 \ 12 \end{pmatrix}$.
The resulting vector $\begin{pmatrix} 8 \ 12 \end{pmatrix}$ points in the same direction as $\begin{pmatrix} 2 \ 3 \end{pmatrix}$ but is four times longer.
Let vector $\vec{b} = \begin{pmatrix} -1 \ 5 \end{pmatrix}$ and scalar $s = -2$.
$-2 \vec{b} = -2 \begin{pmatrix} -1 \ 5 \end{pmatrix} = \begin{pmatrix} -2 \cdot (-1) \ -2 \cdot 5 \end{pmatrix} = \begin{pmatrix} 2 \ -10 \end{pmatrix}$.
The resulting vector $\begin{pmatrix} 2 \ -10 \end{pmatrix}$ is twice as long as $\begin{pmatrix} -1 \ 5 \end{pmatrix}$ and points in the exact opposite direction.
Practical Insights
Scalar multiplication is used extensively:
- Scaling Quantities: If a car's velocity vector is $\vec{v}$, doubling its speed in the same direction means its new velocity is $2\vec{v}$.
- Changing Units: Converting a velocity from meters per second to kilometers per hour involves multiplying by a scalar conversion factor.
- Vector Components: Breaking down a vector into its horizontal and vertical components often involves scalar multiplication with unit vectors.
- Transformations: In computer graphics, scaling objects (making them bigger or smaller) can be represented using scalar multiplication of their position vectors.
In essence, scalar multiplication allows us to stretch, compress, or reverse the direction of a vector while preserving its fundamental nature as a directional quantity. The outcome is always another vector, scaled accordingly.
