Scalar and Vector Quantities
Scalar: A quantity that possesses only magnitude (size).
Examples: mass, time, density, temperature, length, volume, speed, work.
Vector: A quantity that possesses both magnitude and direction, and obeys vector algebra laws.
Examples: displacement, velocity, acceleration, impulse, thrust, torque, angular momentum, angular velocity, weight, force, momentum, electric and magnetic fields.
Remember: Scalars have Size only, Vectors have Value (magnitude) and Virection.
If a quantity answers “how much?” it’s likely scalar. If it answers “how much and which way?” it’s a vector.
Vector Properties and Types
Magnitude (Norm): Length of vector a denoted as |a|.
Unit Vector: Vector with magnitude 1. Denoted as â = a/|a|.
Null Vector (Zero Vector): Vector with zero magnitude, same initial and terminal points.
Equal Vectors: Two vectors with same magnitude and direction.
Parallel Vectors: a and b are parallel if a = kb where k is scalar.
Like Vectors: Angle between them is 0° (same direction).
Unlike Vectors: Angle between them is 180° (opposite direction).
Coplanar Vectors: Vectors lying in the same plane.
Parallel vectors are scalar multiples of each other. If a = kb, they’re parallel (same direction if k>0, opposite if k<0).
Zero vector is unique – it has arbitrary direction because it has no magnitude.
Vector Operations
Addition (Parallelogram Law): a + b = diagonal of parallelogram with adjacent sides a and b.
Subtraction: a – b = a + (-b).
Properties:
- Commutative: a + b = b + a
- Associative: (a + b) + c = a + (b + c)
- Additive Identity: a + 0 = a
- Additive Inverse: a + (-a) = 0
Scalar Multiplication: ma scales the vector by factor m.
For subtraction, think “add the opposite”. Vector subtraction is not commutative: a – b ≠ b – a (they’re negatives of each other).
Parallelogram law: The sum of vectors is the diagonal from the common starting point.
Dot Product (Scalar Product)
a · b = |a||b|cosθ, where θ is angle between vectors.
Properties:
- Commutative: a · b = b · a
- Distributive: a · (b + c) = a · b + a · c
- a · a = |a|²
- If a ⊥ b, then a · b = 0
Projection: Projection of a on b = (a · b)/|b|.
Work Done: W = F · d (force dot displacement).
Dot product gives a scalar result. If vectors are perpendicular, cos90°=0, so dot product is zero.
Remember: Dot product measures how much one vector goes in the direction of another.
Cross Product (Vector Product)
a × b = |a||b|sinθ n, where n is unit vector perpendicular to both.
Properties:
- Anti-commutative: a × b = -(b × a)
- Distributive: a × (b + c) = a × b + a × c
- If a ∥ b, then a × b = 0
- a × a = 0
Geometric Meaning: |a × b| = area of parallelogram with sides a and b.
Moment/Torque: τ = r × F (position vector cross force).
Cross product gives a vector perpendicular to both original vectors. Use right-hand rule for direction.
If vectors are parallel, sin0°=0, so cross product is zero vector.
Remember: Cross product measures “area” spanned by two vectors.
Triple Products
Scalar Triple Product: [a b c] = a · (b × c)
Properties:
- Cyclic permutation doesn’t change value: [a b c] = [b c a] = [c a b]
- Anti-cyclic permutation changes sign: [a b c] = -[a c b]
- Volume of parallelepiped = |[a b c]|
- Vectors are coplanar if [a b c] = 0
Vector Triple Product: a × (b × c) = (a · c)b – (a · b)c
This vector lies in the plane of b and c.
For scalar triple product: “BAC-CAB” rule doesn’t apply here, that’s for vector triple product.
Scalar triple product = 0 means vectors are coplanar (no volume).
For vector triple product, remember: a × (b × c) = (a·c)b – (a·b)c (BAC-CAB rule).
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Answer Key
Study Guidelines for Vector Algebra
Follow these guidelines to master vector algebra effectively:
1. Build Strong Foundations
- Clearly distinguish between scalar and vector quantities from day one
- Master vector representation: magnitude, direction, and notation
- Understand the geometric interpretation of vectors (arrows in space)
2. Visualize Vector Operations
- Draw vectors for addition/subtraction using triangle/parallelogram laws
- Visualize dot product as projection of one vector onto another
- Use right-hand rule for cross product direction visualization
3. Memorize Key Formulas
- Dot product: a·b = |a||b|cosθ
- Cross product magnitude: |a×b| = |a||b|sinθ
- Scalar triple product: [a b c] = a·(b×c)
- Vector triple product: a×(b×c) = (a·c)b – (a·b)c
4. Practice Problem Solving
- Solve problems using both geometric and algebraic approaches
- Apply vectors to physics problems (forces, velocities, etc.)
- Practice with the interactive quiz in this guide
5. Common Pitfalls to Avoid
- Don’t confuse dot product (scalar) with cross product (vector)
- Remember cross product is anti-commutative: a×b = –b×a
- Zero vector is unique: it has no direction, only zero magnitude
- Parallel vectors are scalar multiples, not necessarily equal
6. Exam Preparation Tips
- Focus on conceptual understanding rather than rote memorization
- Practice visualizing 3D vector operations
- Learn to quickly identify when vectors are parallel/perpendicular
- Use the tips and memory aids provided throughout this guide
Pro Tip: Create flashcards for different vector types (unit, null, equal, parallel) and operations. Regularly test yourself with the quiz section.