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Scalars and Vectors for Motion in a Plane (Class 11)

A scalar has only size; a vector has size and direction. Vectors are equal if their size and direction match. Multiplying by a number changes the length (a negative number flips it). Vectors add tail-to-head (triangle or parallelogram law); A − B = A + (−B). Any vector in a plane is A = Ax î + Ay ĵ with Ax = A cos θ, Ay = A sin θ. A·B = AB cos θ is a scalar; A×B has size AB sin θ and is perpendicular to both.

🎬 Step-by-step story

  1. A scalar has only size, like 5 kg. A vector has size and direction. The position vector r goes from O to a point; displacement Δr is the arrow from P to Q.
  2. Equal vectors have the same length and direction, wherever they sit. Multiplying by 2 doubles the length, by −1 flips it, by ½ halves it.
  3. To add, put B's tail on A's head. A + B joins the first tail to the last head. To subtract, add the flipped vector: A − B = A + (−B).
  4. Unit vectors î and ĵ have length 1. Any vector splits into a part along x and a part along y: Ax = A cos θ, Ay = A sin θ.
  5. Dot product A·B = AB cos θ gives a number. Cross product A×B has size AB sin θ, the parallelogram's area, and points out of the plane.
  6. Your turn: change A and B with the sliders. Watch the sum, the difference, the dot product and the cross product change together.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

How is a position vector different from a displacement vector?

Position always starts at the origin O. Displacement starts at the old position and ends at the new one; it does not care where O is.

If I move an arrow, is it a new vector?

No. Sliding it without turning or stretching keeps the same vector. Both red arrows in step 2 are equal.

Why can't I just add the sizes 3 + 4?

Only when they point the same way. At an angle, the tail-to-head arrow is shorter than the two together, as the triangle shows.

Why is Ax = A cos θ and not A sin θ?

Ax is the side next to θ in the right triangle (adjacent), and adjacent/hypotenuse = cos θ. Watch the blue part along x as the arrow turns.

Why is the dot product a number but the cross product a vector?

The dot product only measures how much of B lies along A. The cross product is an area with a direction, so it points out of the plane.

When is A×B zero?

When A and B are parallel or opposite (θ = 0° or 180°): the parallelogram is flat. Put both angles equal in the free play.

Scalars and vectors

A scalar is fully described by a number and a unit: mass, time, temperature, speed, work. A vector needs a size (magnitude) and a direction, and adds by the triangle law: displacement, velocity, acceleration, force.

We write a vector in bold or with an arrow: A or A⃗; its size is |A| or A. Not everything with a direction is a vector: electric current has a direction but adds like a scalar.

Position and displacement vectors

The position vector r of a point P is the arrow from the origin O to P: r = x î + y ĵ.

If an object moves from P (r₁) to Q (r₂), its displacement vector is Δr = r₂ − r₁. It depends only on the start and end points, not on the path taken.

Equal vectors

Two vectors are equal if they have the same magnitude and the same direction. You can slide a vector parallel to itself anywhere and it stays the same vector. Vectors of the same size but opposite direction are negative of each other.

Multiplying a vector by a number

λA is a vector λ times as long as A. If λ > 0 it points the same way; if λ < 0 it points the opposite way; if λ = 0 we get the zero (null) vector. Multiplying by a quantity with units changes the physical meaning: velocity × time = displacement.

Adding and subtracting vectors

Triangle law

Place the tail of B at the head of A. The arrow from A's tail to B's head is A + B.

Parallelogram law

Draw A and B from one point and complete the parallelogram. The diagonal from that point is the resultant R, with R² = A² + B² + 2AB cos θ and tan α = B sin θ/(A + B cos θ).

Addition is commutative (A + B = B + A) and associative. Subtraction: A − B = A + (−B), and |A − B|² = A² + B² − 2AB cos θ.

Unit vector

A unit vector has magnitude 1 and only shows a direction: Â = A/|A|. î, ĵ, k̂ point along the x, y and z axes. So A = |A| Â.

Resolving a vector into rectangular components

If A makes angle θ with the x-axis: Ax = A cos θ, Ay = A sin θ, and A = Ax î + Ay ĵ. Back again: A = √(Ax² + Ay²), tan θ = Ay/Ax.

Adding with components is easy: add the x parts, add the y parts. R = (Ax + Bx) î + (Ay + By) ĵ.

Scalar and vector product

Scalar (dot) product

A·B = AB cos θ = AxBx + AyBy + AzBz. It is a scalar. î·î = 1, î·ĵ = 0. Example: work W = F·s.

Vector (cross) product

|A×B| = AB sin θ, the area of the parallelogram made by A and B. Its direction is perpendicular to both, by the right-hand rule. î×ĵ = k̂, ĵ×k̂ = î, k̂×î = ĵ, and B×A = −A×B. Example: torque τ = r×F.

Key formulas and definitions

Worked examples

1. Two forces of 3 N and 4 N act at a right angle. Find the resultant.

Step 1: θ = 90°, cos 90° = 0. Step 2: R = √(9 + 16 + 0) = 5 N. Step 3: tan α = 4/3 → α ≈ 53° from the 3 N force.

2. A particle moves from P(1, 2) m to Q(4, 6) m. Find its displacement vector and its size.

Step 1: Δr = (4 − 1) î + (6 − 2) ĵ = 3î + 4ĵ m. Step 2: |Δr| = √(9 + 16) = 5 m.

3. A velocity of 20 m/s makes 30° with the x-axis. Find its components.

Step 1: vx = 20 cos 30° = 20 × 0.866 = 17.3 m/s. Step 2: vy = 20 sin 30° = 10 m/s. Step 3: v = 17.3 î + 10 ĵ m/s.

4. Find the unit vector along A = 6î − 8ĵ.

Step 1: |A| = √(36 + 64) = 10. Step 2: Â = A/|A| = 0.6î − 0.8ĵ. Step 3: check: 0.36 + 0.64 = 1 ✓.

5. Two vectors of size 5 each make 60° with each other. Find |A + B| and |A − B|.

Step 1: |A + B|² = 25 + 25 + 2·25·cos 60° = 75 → |A + B| ≈ 8.66. Step 2: |A − B|² = 25 + 25 − 25 = 25 → |A − B| = 5.

6. A = 2î + 3ĵ and B = 4î − ĵ. Find A·B and the angle between them.

Step 1: A·B = 2·4 + 3·(−1) = 5. Step 2: |A| = √13 ≈ 3.61, |B| = √17 ≈ 4.12. Step 3: cos θ = 5/(3.61 × 4.12) ≈ 0.336 → θ ≈ 70.4°.

7. For the same A = 2î + 3ĵ and B = 4î − ĵ, find A×B and the area of their parallelogram.

Step 1: A×B = (AxBy − AyBx) k̂ = (2·(−1) − 3·4) k̂ = −14 k̂. Step 2: area = |A×B| = 14 square units. Step 3: the minus sign means it points into the plane.

8. A force F = 3î + 4ĵ N moves a box by s = 5î m. Find the work done.

Step 1: W = F·s = 3·5 + 4·0. Step 2: W = 15 J. Only the part of F along the motion does work.

Common mistakes

Practice quiz

1. Which of these is a vector?
2. The magnitude of a unit vector is:
3. If A·B = 0 for non-zero vectors, the angle between them is:
4. î × ĵ equals:
5. The largest possible size of A + B is:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What is the difference between scalars and vectors?

A scalar has only magnitude (mass, time). A vector has magnitude and direction and adds by the triangle law (velocity, force).

What is the resolution of a vector?

Splitting a vector into perpendicular parts: Ax = A cos θ along x and Ay = A sin θ along y.

What is the difference between the dot product and the cross product?

The dot product AB cos θ gives a scalar (like work). The cross product has size AB sin θ and a direction perpendicular to both (like torque).

Where this is taught

Canada (Ontario)Grade 12D. Applications of Geometry
ItalySecondaria di secondo grado – classe 1ªArithmetic and algebra
ItalySecondaria di secondo grado – classe 2ªArithmetic and algebra
Ukraine9 класVectors on the plane
CBSE (India)Class 11Kinematics
USA (Common Core, NGSS, AP)Grade 11Kinematics
South Korea고등학교 2학년Text data
South Korea고등학교 2학년Space-time and motion
South Korea고등학교 3학년Mechanical interactions
Russia9 классMechanical phenomena
China高一Ch.6 Plane vectors
China高一Compulsory 1 Ch.3 Forces

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