When a vector is multiplied by a scalar (a single number), the result is another vector. The effect of this operation on the original vector's direction depends on the value of the scalar being used.
Understanding Scalar Multiplication
Scalar multiplication is one of the fundamental operations involving vectors. It involves taking a vector, such as $\vec{v}$, and multiplying it by a real number, $k$.
- The result of multiplying a vector by a scalar is a vector quantity. As stated in the reference, "The result of multiplying a vector by a scalar is a vector quantity."
- This new vector is parallel to the original vector, but its magnitude and potentially its direction are altered.
How Scalars Affect Vector Direction
The direction of the resulting vector is determined entirely by the sign of the scalar multiplier.
-
Positive Scalar (k > 0): If you multiply a vector by a positive scalar, the direction of the vector remains exactly the same. Only its magnitude changes. Based on the provided reference, "The vector has the same direction," which specifically applies when the scalar is positive.
- If the scalar is greater than 1 (k > 1), the vector becomes longer (magnitude increases).
- If the scalar is between 0 and 1 (0 < k < 1), the vector becomes shorter (magnitude decreases).
- If the scalar is exactly 1 (k = 1), the vector remains unchanged ($1 \times \vec{v} = \vec{v}$).
-
Negative Scalar (k < 0): If you multiply a vector by a negative scalar, the direction of the vector is reversed by 180 degrees. The resulting vector points in the opposite direction to the original vector. Its magnitude changes based on the absolute value of the scalar.
- For example, multiplying by -1 ($\text{-}1 \times \vec{v} = \text{-}\vec{v}$) results in a vector with the same magnitude but pointing in the exact opposite direction.
- Multiplying by -2 ($\text{-}2 \times \vec{v}$) results in a vector pointing in the opposite direction with twice the magnitude.
-
Scalar is Zero (k = 0): If you multiply a vector by zero, the result is the zero vector ($\vec{0}$). The zero vector has zero magnitude and is considered to have no specific direction.
Visualizing the Change
Imagine a vector as an arrow pointing from one point to another.
- Multiplying by a positive scalar like 2 just makes the arrow twice as long but still pointing the same way.
- Multiplying by a negative scalar like -1 keeps the arrow the same length but flips it around to point the opposite way.
- Multiplying by zero shrinks the arrow down to a single point with no length or direction.
Summary of Effects
Here's a quick look at how different types of scalars affect a vector's direction and magnitude:
| Scalar Type | Effect on Direction | Effect on Magnitude |
|---|---|---|
| Positive (k > 0) | Stays the same | Scaled by k |
| Negative (k < 0) | Reverses (180° change) | Scaled by the absolute value of k ( |
| Zero (k = 0) | Becomes the zero vector (no direction) | Becomes zero |
In essence, scalar multiplication scales the vector, and only negative scalars introduce a change in its directional orientation.
