Introduction:

In physics, vectors are fundamental in representing quantities that have both magnitude and direction, such as displacement, velocity, and force. Understanding how to manipulate vectors, especially through multiplication by real numbers, is crucial for analyzing various physical phenomena. This process, known as scalar multiplication, alters the magnitude of a vector without changing its direction, providing a foundation for more complex vector operations.

Multiplication of Vectors by Real Numbers:

Scalar Multiplication:

Scalar multiplication involves multiplying a vector by a real number (scalar). If A is a vector and λ is a scalar, the product λA results in a new vector whose magnitude is |λ| times the magnitude of A, and whose direction depends on the sign of λ:

• If λ > 0: The direction of λA is the same as A.

• If λ < 0: The direction of λA is opposite to that of A.

The magnitude of the new vector is given by:

|λA| = |λ| |A|

Effects on Magnitude and Direction:

• Magnitude: The magnitude of the vector is scaled by the absolute value of the scalar. For instance, if λ = 2, the magnitude of the vector doubles; if λ = 0.5, the magnitude is halved.

• Direction: The direction remains unchanged if λ is positive. If λ is negative, the direction reverses.

Geometric Interpretation:

Geometrically, scalar multiplication stretches or compresses the vector by the factor |λ|. A positive λ maintains the original direction, while a negative λ reflects the vector across the origin.

Examples:

Doubling a Vector:

Let A be a vector with magnitude 5 units in the positive x-direction. If λ = 2:

λA = 2A

• Magnitude: |2| × 5 = 10 units

• Direction: Same as A (positive x-direction)

Reversing a Vector:

If λ = -1:

λA = -1 × A = -A

• Magnitude: |−1| × 5 = 5 units

• Direction: Opposite to A (negative x-direction)

Halving a Vector:

If λ = 0.5:

λA = 0.5A

• Magnitude: |0.5| × 5 = 2.5 units