1. Meaning of Scalar Multiplication
Scalar multiplication means multiplying a vector by a real number (called a scalar). This operation changes the magnitude of the vector but keeps its direction the same, unless the scalar is negative.
If \(k\) is a scalar and \(\vec{v}\) is a vector, then:
\[ k\vec{v} = k \cdot \langle a, b, c \rangle = \langle ka, kb, kc \rangle \]
This stretches, shrinks, or reverses the vector depending on the value of \(k\).
2. Effect of a Positive Scalar
Multiplying a vector by a positive scalar keeps the direction the same. The magnitude scales by the same factor.
If \(k > 0\), then \(k\vec{v}\) points in the same direction as \(\vec{v}\).
2.1. Example
For \(\vec{v} = \langle 2, 3 \rangle\) and \(k = 3\):
\[ 3\vec{v} = \langle 6, 9 \rangle \]
The new vector is longer but points the same way.
3. Effect of a Negative Scalar
Multiplying by a negative scalar reverses the direction of the vector. The magnitude still scales by the absolute value of the scalar.
If \(k < 0\), then \(k\vec{v}\) points in the opposite direction of \(\vec{v}\).
3.1. Example
For \(\vec{v} = \langle 1, -2 \rangle\) and \(k = -2\):
\[ -2\vec{v} = \langle -2, 4 \rangle \]
The vector now points opposite to the original direction.
4. Effect of a Scalar Between 0 and 1
When \(0 < k < 1\), the vector gets shorter but keeps its direction.
This is like compressing the vector while preserving the direction.
4.1. Example
If \(\vec{v} = \langle 10, 0 \rangle\) and \(k = 0.2\):
\[ 0.2\vec{v} = \langle 2, 0 \rangle \]
The vector is shorter but still on the positive x-axis.
5. Effect of Zero Scalar
If the scalar is zero, the result is the zero vector, no matter what the original vector was.
\[ 0 \cdot \vec{v} = \vec{0} \]
5.1. Example
For any vector \(\vec{v} = \langle a, b, c \rangle\):
\[ 0\vec{v} = \langle 0, 0, 0 \rangle \]
6. Scalar Multiplication in Component Form
Scalar multiplication is applied separately to each component of a vector.
In 2D:
\[ k\langle a, b \rangle = \langle ka, kb \rangle \]
In 3D:
\[ k\langle a, b, c \rangle = \langle ka, kb, kc \rangle \]
6.1. Example
Let \(\vec{v} = \langle 3, -1, 4 \rangle\) and \(k = 2\):
\[ 2\vec{v} = \langle 6, -2, 8 \rangle \]
7. Using Scalar Multiplication to Form Unit Vectors
A vector can be converted into a unit vector by multiplying it by the reciprocal of its magnitude:
\[ \hat{v} = \dfrac{1}{|\vec{v}|}\vec{v} \]
This shows how scalar multiplication helps normalize vectors for direction-only use.
7.1. Example
If \(\vec{v} = \langle 6, 8 \rangle\), then:
\[ |\vec{v}| = 10 \]
\[ \hat{v} = \dfrac{1}{10}\langle 6, 8 \rangle = \langle 0.6, 0.8 \rangle \]