Multiplication of Vectors by Real Numbers
Key Concepts
- When you multiply a vector 𝐀 by a positive number λ, the direction stays the same, but the magnitude (length) changes by a factor of λ:
|λ𝐀| = λ|𝐀| (if λ > 0).
Example: Doubling vector 𝐀 gives 2𝐀, which is twice as long but points the same way. - Multiplying by a negative number -λ flips the direction and scales the magnitude by λ:
Example: -1.5𝐀 is 1.5 times as long as 𝐀 but points in the opposite direction. - The factor λ can have units (like time or speed). The units of λ𝐀 will be the product of the units of λ and 𝐀.
Example: Multiplying a velocity vector (m/s) by time (s) gives a displacement vector (m).
Visualizing Vector Multiplication
- Positive scaling (λ > 0): Stretches or shrinks the vector without changing its direction.
- Negative scaling (λ < 0): Reverses the direction and adjusts the length.
NEET High-Yield Ideas
- Scalar Multiplication: How multiplying a vector by a scalar (positive/negative) affects its magnitude and direction.
- Direction Matters: Even if two vectors have the same length, they’re unequal if their directions differ.
- Units in Physics: Multiplying vectors by scalars with units (e.g., time × velocity = displacement) is common in kinematics.
- Graphical Representation: Drawing vectors before/after scaling helps visualize concepts like force, velocity, or displacement.
Example Problems (Conceptual)
- If vector 𝐀 represents a 5 N force eastward, what does -2𝐀 represent?
Answer: A 10 N force westward (direction flipped, magnitude doubled). - How does multiplying a displacement vector by time affect its units?
Answer: Displacement (meters) × time (seconds) = meters × seconds (not standard; likely a mistake—usually, velocity × time = displacement).
Remember!
- Vectors are more than just length—direction is key!
- Negative multipliers “flip” the vector like a mirror image.
