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Scalar Product or Dot Product
The scalar product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them. A⃗ · B⃗ = AB cos θ. The result is a scalar.
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Recommended for Class 11
Subtopic 1 — Scalar Product or Dot Product
Scalar Product (Dot Product)
Terms Used in the Formula
In: A⃗ · B⃗ = AB cos θ A is the magnitude of A⃗. B is the magnitude of B⃗. θ is the smaller angle between the vectors. The dot (·) represents scalar multiplication. The answer is a scalar quantity.
Physical Meaning of Dot Product
The dot product measures the component of one vector acting along the direction of another vector. The component of B⃗ along A⃗ is: B cos θ Therefore: A⃗ · B⃗ = A(B cos θ) Similarly: A⃗ · B⃗ = B(A cos θ) A larger dot product indicates that the vectors act more strongly in the same direction.
Dot Product at Special Angles
Parallel Vectors When θ = 0°: A⃗ · B⃗ = AB cos 0° = AB The dot product has its maximum positive value. Perpendicular Vectors When θ = 90°: A⃗ · B⃗ = AB cos 90° = 0 Therefore, the dot product of two perpendicular vectors is zero. Opposite Vectors When θ = 180°: A⃗ · B⃗ = AB cos 180° = −AB The dot product has its maximum negative value.
Dot Product Using Components
Consider: A⃗ = Aₓ î + Aᵧ ĵ + Az k̂ and: B⃗ = Bₓ î + Bᵧ ĵ + Bz k̂ Their dot product is: A⃗ · B⃗ = AₓBₓ + AᵧBᵧ + AzBz For two-dimensional vectors: A⃗ · B⃗ = AₓBₓ + AᵧBᵧ
Dot Products of Unit Vectors
A unit vector has a magnitude of one. Therefore: î · î = 1 ĵ · ĵ = 1 k̂ · k̂ = 1 Different Cartesian unit vectors are perpendicular. Therefore: î · ĵ = 0 ĵ · k̂ = 0 k̂ · î = 0
Properties of Dot Product
Commutative Property A⃗ · B⃗ = B⃗ · A⃗ Distributive Property A⃗ · (B⃗ + C⃗) = A⃗ · B⃗ + A⃗ · C⃗ Multiplication of a Vector by Itself A⃗ · A⃗ = A² Therefore: A = √(A⃗ · A⃗) Scalar Multiplication (mA⃗) · (nB⃗) = mn(A⃗ · B⃗)
Angle Between Two Vectors
From: A⃗ · B⃗ = AB cos θ We get: cos θ = (A⃗ · B⃗) / AB Therefore: θ = cos⁻¹((A⃗ · B⃗) / AB) If the dot product is: Positive, the angle is less than 90°. Zero, the vectors are perpendicular. Negative, the angle is greater than 90°.
Applications of Dot Product
The dot product is used to calculate
- Work Done: W = F⃗ · s⃗ = Fs cos θ
- Instantaneous Power: P = F⃗ · v⃗
- Component of One Vector Along Another: A∥ = A cos θ
- Angle Between Two Vectors: θ = cos⁻¹((A⃗ · B⃗) / AB)
Solved Examples — Dot Product
Solved Example 1: Dot Product Using Magnitudes
Two vectors have magnitudes 6 and 8. The angle between them is 60°. Calculate their dot product.
- A = 6
- B = 8
- θ = 60°
A⃗ · B⃗ = AB cos θ
A⃗ · B⃗ = (6)(8) cos 60°
A⃗ · B⃗ = 48(0.5)
A⃗ · B⃗ = 24
Final answer: A⃗ · B⃗ = 24
Solved Example 2: Dot Product Using Components
Calculate the dot product of A⃗ = 2î − ĵ + 3k̂ and B⃗ = 4î + 2ĵ − 2k̂.
A⃗ · B⃗ = AₓBₓ + AᵧBᵧ + AzBz
A⃗ · B⃗ = (2)(4) + (−1)(2) + (3)(−2)
A⃗ · B⃗ = 8 − 2 − 6
A⃗ · B⃗ = 0
Key Step: A zero dot product means that the two non-zero vectors are perpendicular.
Final answer: A⃗ · B⃗ = 0
Solved Example 3: Work Done
A force of 50 N moves an object through 4 m. The angle between force and displacement is 60°. Find the work done.
- F = 50 N
- s = 4 m
- θ = 60°
W = Fs cos θ
W = (50)(4) cos 60°
W = 200(0.5)
W = 100 J
Final answer: W = 100 J
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Practice
Beginner
Q1. Define the dot product.
Beginner
Q2. Write the dot products of Cartesian unit vectors.
Beginner
Q3. Write the cross products of Cartesian unit vectors.
Intermediate
Q4. Calculate the dot product of vectors having magnitudes 10 and 12 with an angle of 60°.
Intermediate
Q5. Find A⃗ · B⃗ if A⃗ = 3î + 4ĵ and B⃗ = 2î − ĵ.
Intermediate
Q6. Determine whether A⃗ = 2î + 3ĵ and B⃗ = 3î − 2ĵ are perpendicular.
Intermediate
Q7. Calculate work done by a 30 N force over 5 m at an angle of 60°.
Advanced
Q8. Explain why reversing the order changes the cross product but not the dot product.
Exam Practice
Important Short Question
Q1. Define the scalar product.
Important Short Question
Q2. When is the dot product zero?
Long Question Preparation
Q3. Explain the dot product and cross product of two vectors.