Skip to lesson

Physics · Class 11

Head-to-tail vectors

Motion in a Plane

Move the tail of the second arrow to the head of the first. The resultant joins the original tail to the final head. Subtracting B means adding −B. A negative scale factor reverses direction; its absolute value scales length. Equal vectors have equal components, regardless of where they are drawn.

Open the interactive experiment →

Join the arrows to find the resultant.

இரண்டாவது திசையனின் தொடக்கத்தை முதல் திசையனின் முடிவில் வைக்கவும். தொடக்கத்திலிருந்து இறுதி முடிவுக்குச் செல்லும் திசையன் கூட்டுத்திசையன். கழித்தல் என்பது எதிர்த் திசையனைச் சேர்ப்பதாகும்.

Key notes, relationships and model limits

Key relationship

R = A ± kB; Rₓ = Aₓ ± kBₓ; Rᵧ = Aᵧ ± kBᵧ. |kB| = |k| |B|.

Remember

Move the tail of the second arrow to the head of the first. The resultant joins the original tail to the final head. Subtracting B means adding −B. A negative scale factor reverses direction; its absolute value scales length. Equal vectors have equal components, regardless of where they are drawn.

Common trap

The magnitude of a vector sum is not generally the sum of the magnitudes.

Model limits

All arrows represent displacements in metres on fixed orthogonal axes. A free vector can be translated without changing it. Negative k reverses B. Equality readout compares the original A and B before scaling.

What you will learn

Add and subtract components; interpret positive and negative scaling; distinguish equality from equal length.

Try the experiment

Try the perpendicular preset and check the resultant length. Then subtract equal arrows and use a negative scale factor.

Think of it this way

Go three steps east, then four steps north. The direct trip to the same destination is five steps northeast.

Worked example

A − 2B with A=(4,1) m and B=(1,3) m gives R=(4−2, 1−6)=(2,−5) m. Its length is √29 m.

Why this matters

Head-to-tail construction and component addition are two representations of the same operation.

Before you start

Rectangular axes · Pythagoras · sine and cosine

Check your understanding

A = 3î m and B = 4ĵ m. What is |A + B|?

Read the explanation

The components are (3,4) m. Pythagoras gives √(3²+4²) = 5 m. Adding the lengths would describe the walked path, not the displacement.

Course connection: CBSE XI Motion in a Plane; NEET/JEE kinematics. Use the assumptions stated in each question.