Vectors — Addition, Dot Product and Cross Product
A vector has a size and a direction. That is the whole definition, and it is why 3 N east plus 4 N north is 5 N north-east and not 7 N of anything. Class 11 physics opens with vectors because almost nothing after it works without them: force, velocity, acceleration, momentum, electric field and magnetic field are all vectors. This article covers the three operations that carry the most marks — addition, the dot product and the cross product — and ends with an identity that lets you check a cross product without redoing it.
The formulas
Magnitude: |A| = √(Ax² + Ay² + Az²)
Addition: A + B = (Ax+Bx)i + (Ay+By)j + (Az+Bz)k
Parallelogram law: R = √(A² + B² + 2AB cosθ)
Dot product: A · B = |A||B| cosθ = AxBx + AyBy + AzBz
Cross product magnitude: |A × B| = |A||B| sinθ
Dot or cross? Decide by the answer you need
| Dot product A · B | Cross product A × B | |
|---|---|---|
| Answer is a | Number (scalar) | Vector |
| Uses | cosθ | sinθ |
| Maximum when | θ = 0° (parallel) | θ = 90° (perpendicular) |
| Zero when | θ = 90° (perpendicular) | θ = 0° or 180° (parallel) |
| Order matters? | No: A · B = B · A | Yes: A × B = −(B × A) |
| Physics examples | Work W = F · s · magnetic flux Φ = B · A | Torque τ = r × F · magnetic force F = qv × B · angular momentum L = r × p |
A quick way to remember it: the dot product asks "how much of A points along B?" and the cross product asks "how much of A is at right angles to B?" — and returns a vector perpendicular to both.
Worked example 1 — addition in two dimensions
Question: A = 3i + 4j, B = i − 2j. Find A + B, its magnitude and its direction.
Add component by component:
x: 3 + 1 = 4 y: 4 + (−2) = 2
A + B = 4i + 2j
Magnitude: |A + B| = √(4² + 2²) = √(16 + 4) = √20 = 4.472
Direction: tanα = 2 ÷ 4 = 0.5, so α = 26.57° above the x-axis.
Check with the parallelogram law. |A| = √(9+16) = 5;
|B| = √(1+4) = √5 = 2.2361.
A · B = 3(1) + 4(−2) = 3 − 8 = −5
cosθ = −5 ÷ (5 × 2.2361) = −5 ÷ 11.1803 =
−0.44721
R = √(25 + 5 + 2 × 5 × 2.2361 × (−0.44721))
R = √(30 + 22.3607 × (−0.44721)) = √(30 − 10.000) =
√20 = 4.472. ✔
Worked example 2 — two forces at an angle
Question: Forces of 6 N and 8 N act at a point with 60° between them. Find the magnitude and direction of the resultant.
Parallelogram law:
R = √(6² + 8² + 2 × 6 × 8 × cos 60°)
R = √(36 + 64 + 96 × 0.5) = √(100 + 48) = √148 =
12.17 N
Check by components. Put the 6 N force along the x-axis, so it is
(6, 0). The 8 N force at 60° is (8 cos 60°, 8 sin 60°) = (4.000, 6.9282).
Sum = (6 + 4, 0 + 6.9282) = (10, 6.9282)
|R| = √(100 + 48.000) = √148 = 12.17 N. ✔
Direction: tanα = 6.9282 ÷ 10 = 0.69282, so α = 34.7° measured from the 6 N force.
Sanity check: the answer must lie between 8 − 6 = 2 N (forces opposite) and 8 + 6 = 14 N (forces aligned). 12.17 N sits inside that range, closer to 14 because 60° is a fairly small angle.
Worked example 3 — dot product and the angle between two vectors
Question: A = 2i + 3j − k and B = 4i − j + 2k. Find A · B and the angle between them.
Dot product:
A · B = (2)(4) + (3)(−1) + (−1)(2) = 8 − 3 − 2 =
3
Magnitudes:
|A| = √(4 + 9 + 1) = √14 = 3.7417
|B| = √(16 + 1 + 4) = √21 = 4.5826
Angle: cosθ = 3 ÷ (3.7417 × 4.5826) = 3 ÷
17.1464 = 0.17496
θ = 79.9°
The dot product is small and positive, so the vectors are nearly — but not quite — perpendicular. If it had come out exactly 0, they would be at 90°; a negative value would mean an obtuse angle.
Worked example 4 — cross product, checked three ways
Question: For the same A and B, find A × B and its magnitude.
Set out the determinant with A in the middle row:
| i | j | k |
|---|---|---|
| 2 | 3 | −1 |
| 4 | −1 | 2 |
i component: (3)(2) − (−1)(−1) = 6 − 1 = 5
j component: −[(2)(2) − (−1)(4)] = −[4 + 4] =
−8 (note the minus sign in front of the j term)
k component: (2)(−1) − (3)(4) = −2 − 12 =
−14
A × B = 5i − 8j − 14k
Check 1 — it must be perpendicular to both.
(A × B) · A = 5(2) + (−8)(3) + (−14)(−1) = 10 − 24 + 14
= 0 ✔
(A × B) · B = 5(4) + (−8)(−1) + (−14)(2) = 20 + 8 − 28
= 0 ✔
Magnitude: |A × B| = √(25 + 64 + 196) = √285 = 16.88
Check 2 — against |A||B| sinθ. From example 3,
cosθ = 0.17496, so sinθ = √(1 − 0.17496²) =
√(1 − 0.030611) = √0.969389 = 0.98458.
|A||B| sinθ = 17.1464 × 0.98458 = 16.88 ✔
Check 3 — the identity worth memorising:
|A × B|² + (A · B)² = |A|²|B|²
285 + 3² = 285 + 9 = 294, and |A|²|B|² = 14 × 21 = 294. ✔
That third check takes about ten seconds and catches almost every sign error people make in the j term. Use it every time.
Common mistakes that cost marks
- Adding magnitudes instead of vectors. 3 N and 4 N at right angles give 5 N, not 7 N. Only when the vectors are parallel do the magnitudes simply add.
- Forgetting the minus sign on the j term of a cross product. It is the single most common slip in this topic, and check 3 above catches it immediately.
- Writing a vector answer for a dot product. A · B is a plain number. If your answer has an i in it, you did a cross product.
- Using sin in the parallelogram law. The resultant formula takes cosθ, and the sign in front of 2AB cosθ is plus when θ is the angle between the vectors. The minus-sign version belongs to the difference A − B.
- Assuming A × B = B × A. It does not; the two differ by a minus sign. This matters physically — reverse the order in τ = r × F and you reverse the direction the object spins.
- Measuring the angle wrongly. θ is the angle between the vectors when both are drawn from the same starting point, not the angle in the head-to-tail triangle.
- Leaving the calculator in radian mode. cos 60 in radians is −0.9524, not 0.5, and the whole answer collapses.
Where vectors appear in exams
| Exam | Typical use |
|---|---|
| CBSE/ICSE Class 11 Physics | Motion in a plane — resolution of vectors, relative velocity, projectile motion |
| CBSE Class 12 Maths | Vector algebra: dot and cross products, projections, area of a triangle and a parallelogram |
| CBSE Class 12 Physics | Torque, magnetic force on a moving charge, magnetic flux, angular momentum |
| JEE Main & Advanced | Scalar triple product, coplanarity, shortest distance between lines, equilibrium of forces |
| IIT-JAM / GATE | Vector calculus foundations — gradient, divergence and curl all start here |
Two geometry results are worth carrying into the exam hall: the area of a parallelogram with sides A and B is |A × B|, and the area of the triangle they form is half of that. In example 4, that gives a parallelogram of area 16.88 and a triangle of area 8.44 — no extra work, just a reading of the answer you already have.
Check your working in seconds. There is no dedicated vector tool in the suite, so this one links to the Scientific Calculator, which is what a plain visit opens. Use it for the square roots, the inverse-tangent and the inverse-cosine steps in the examples above — those are where the arithmetic slips happen.
Open the ABC Chemistry Calculator Suite →Vectors are the first Class 11 topic where weak trigonometry starts to show, and it shows in chemistry too, the moment bond angles and geometry appear. ABC Chemistry runs Class 11–12 coaching at the Gurugram centre and online classes across India — details at abcchemistry.in. For one-to-one help at home in Delhi, Noida or Gurgaon, see delhihometutor.com.