Vector Scaling Fundamentals

Vector scaling is a fundamental operation in physics and mathematics where a vector is multiplied by a scalar (a real number). When a vector, let's say $\vec{A}$, is multiplied by a scalar $\alpha$, the result is a new vector, $\vec{B} = \alpha \vec{A}$. This operation changes the magnitude of the vector and can also change its direction, depending on the value of the scalar $\alpha$.

Impact of Scalar Multiplication on Vector Properties

The scalar $\alpha$ plays a crucial role in determining the new vector's magnitude and direction. Let's explore how:

• Magnitude of the Scaled Vector: The magnitude of the new vector $\vec{B}$ is given by the absolute value of the scalar multiplied by the magnitude of the original vector. Mathematically, this is expressed as $\left| \vec{B} \right| = \left| \alpha \right| \left| \vec{A} \right|$. This means the magnitude of the vector is scaled by the absolute value of the scalar.
• Direction of the Scaled Vector: The direction of the scaled vector depends on the sign of the scalar $\alpha$.
  • If $\alpha > 0$ (positive scalar), the direction of the scaled vector remains the same as the original vector.
  • If $\alpha < 0$ (negative scalar), the direction of the scaled vector reverses, pointing in the exact opposite direction to the original vector.
  • If $\alpha = 0$, the scaled vector becomes a zero vector, which has zero magnitude and an undefined direction.

Analyzing Vector Scaling with $\alpha = -1$

The question specifically asks what happens when the scalar $\alpha = -1$. Let's apply the rules of vector scaling to this specific case.

Consider an original vector $\vec{A}$. When it is multiplied by $\alpha = -1$, the new vector is $\vec{B} = -1 \cdot \vec{A} = - \vec{A}$.

• Magnitude Analysis:

  Using the formula for magnitude: $\left| \vec{B} \right| = \left| \alpha \right| \left| \vec{A} \right|$

  Substitute $\alpha = -1$: $\left| \vec{B} \right| = \left| -1 \right| \left| \vec{A} \right|$

  Since $\left| -1 \right| = 1$, we get: $\left| \vec{B} \right| = 1 \cdot \left| \vec{A} \right| = \left| \vec{A} \right|$

  This shows that the magnitude of the vector remains exactly the same when multiplied by $-1$.

• Direction Analysis:

  According to the rules for direction, if the scalar $\alpha$ is negative ($\alpha < 0$), the direction of the vector reverses.

  In this case, $\alpha = -1$, which is a negative value. Therefore, the direction of the vector will reverse, pointing 180 degrees opposite to its original direction.

Conclusion for $\alpha = -1$ Scaling

When a vector is scaled by $\alpha = -1$, its magnitude remains unchanged, and its direction reverses. This means the vector maintains its length but points in the exact opposite direction.

Let's review the given options:

• Option 1: Magnitude remains same and direction of the vector reverses - This aligns perfectly with our analysis. The magnitude is $\left| \vec{A} \right|$ and the direction is reversed.
• Option 2: Magnitude changes and direction of the vector remains same - This is incorrect. The magnitude does not change, and the direction reverses.
• Option 3: Magnitude and direction of the vector, both remain same - This is incorrect. The direction specifically reverses.
• Option 4: Magnitude becomes zero and direction of the vector reverse - This is incorrect. The magnitude remains the same, not zero. A magnitude of zero only occurs if the scalar is zero or the original vector is a zero vector.

Therefore, the statement "Magnitude remains same and direction of the vector reverses" accurately describes the outcome when a vector is scaled by a scalar of $-1$.