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NEET UG (Medical Entrance) · Physics (JEE & NEET)

Basic Vectors for Physics

Physics-first vector tools: components, unit vectors, dot and cross products, relative velocity, forces, torque, angular momentum and field/flux framing.

A bridge topic for using vectors inside Physics, not a replacement for Math Vector Algebra.

  • NEET UG (Medical Entrance)
  • Medium level
  • 5 concepts
  • 5 practice questions

1Scalars, vectors and components

A scalar is complete with one number; a vector is not. A force of 10\text{ N} needs direction before it can predict motion. Components are how that direction becomes algebra: after choosing axes, A_x=A\cos\theta and A_y=A\sin\theta when the angle is measured from the x-axis.

Figure. The arrow is the vector; the dashed horizontal and vertical legs are the components with signs set by the axes.

How it works

  1. Choose axesPick x and y to simplify the physics, often along a surface or acceleration.
  2. ProjectUse cosine for the component adjacent to the measured angle.
  3. RebuildWrite \vec A=A_x\hat i+A_y\hat j and keep signs.

Components of a 10 N force

A force of 10\text{ N} acts at 30^\circ above the x-axis.

  • F_x=10\cos30^\circ8.66\text{ N}
  • F_y=10\sin30^\circ5.00\text{ N}
  • \vec F8.66\hat i+5.00\hat j\text{ N}

Pro tip. The trigonometric function is decided by where the angle is drawn, not by memorising horizontal = cos.

A vector 5\hat i-12\hat j has magnitude
  1. 7
  2. 13
  3. 17

Magnitude is \sqrt{5^2+(-12)^2}=13.

2Dot product: the along-part

The dot product keeps only the component of one vector along another: \vec A\cdot\vec B=AB\cos\theta. That is why work is \vec F\cdot\vec s, electric flux is \vec E\cdot\vec A, and power can be \vec F\cdot\vec v. A perpendicular force may be large and still do no work.

Figure. The dot product keeps only the along-part: drop a perpendicular from the tip of F onto the displacement line. That shadow is F cos θ, and work is that length times s.

How it works

  1. Find the angleUse the angle between the two vectors, not either vector with an arbitrary axis.
  2. Take the projectionOnly the parallel component contributes.
  3. Read the signPositive helps, zero is perpendicular, negative opposes.

Work by an oblique force

A 20\text{ N} force pulls a block 5\text{ m} while making 60^\circ with displacement.

  • W=Fs\cos\theta20\times5\times\cos60^\circ
  • W50\text{ J}

Pro tip. The 20\text{ N} is not all useful; only 20\cos60^\circ lies along the displacement.

A force is perpendicular to displacement. The work done by that force is
  1. Maximum
  2. Zero
  3. Negative always

W=Fs\cos90^\circ=0.

3Cross product: the perpendicular-effect part

The cross product measures the part of a vector perpendicular to another vector and points normal to their plane. Its magnitude is AB\sin\theta. In Physics this is torque \vec\tau=\vec r\times\vec F, angular momentum \vec L=\vec r\times\vec p and magnetic force q\vec v\times\vec B. The line of action matters as much as the force size.

Figure. Torque and magnetic force both carry |A×B| = AB sin θ. The magnitude peaks when the two vectors are perpendicular and vanishes when they are parallel — the direction itself is out of the plane and named by the right-hand rule in the prose.

How it works

  1. Use perpendicular armrF\sin\theta is force times perpendicular lever arm.
  2. Set directionCurl fingers from the first vector to the second; thumb gives the cross-product direction.
  3. Check zero casesParallel vectors give no torque or magnetic deflection.

Torque from a force

A 10\text{ N} force is applied at the end of a 0.50\text{ m} rod at 30^\circ to the rod.

  • \tau=rF\sin\theta0.50\times10\times\sin30^\circ
  • \tau2.5\text{ N m}

Pro tip. A force along the rod has zero moment arm, so it cannot turn the rod no matter how large it is.

For maximum torque with fixed r and F, the force should be
  1. Parallel to the radius
  2. Perpendicular to the radius
  3. At 30^\circ to the radius

\tau=rF\sin\theta is largest at 90^\circ.

4Relative velocity is frame subtraction

Relative velocity is not a new kind of velocity; it is the same motion described from another moving frame. The velocity of A as seen from B is \vec v_{A/B}=\vec v_A-\vec v_B. This one subtraction underlies rain-man problems, river-boat problems and two-projectile separation.

Figure. Relative velocity is frame subtraction on one line: from B's seat, remove v_B from every ground velocity. The leftover segment is what A does relative to B.

How it works

  1. Name the observerThe denominator in v_{A/B} is the observer frame.
  2. Subtract observer velocityUse \vec v_A-\vec v_B with signs and components.
  3. Solve in that frameA moving observer can turn a two-body problem into one-body motion.

Rain seen by a walker

Rain falls with velocity -10\hat j\text{ m s}^{-1} and a person walks 6\hat i\text{ m s}^{-1}. Find rain velocity relative to the person.

  • \vec v_{rain/person}=\vec v_{rain}-\vec v_{person}-6\hat i-10\hat j
  • Speed\sqrt{6^2+10^2}=11.7\text{ m s}^{-1}

Pro tip. Subtract the observer. The rain appears tilted backward because the walking velocity is subtracted.

If two projectiles have the same acceleration \vec g, their relative acceleration is
  1. \vec g
  2. 2\vec g
  3. 0

\vec a_{A/B}=\vec a_A-\vec a_B=\vec g-\vec g=0.

5Forces, fields and flux use the same vector grammar

Vector notation is shared across mechanics and fields. Concurrent-force equilibrium says the force vectors add to zero, usually by \sum F_x=0 and \sum F_y=0. Electric and magnetic fields point in the direction of force per test charge or pole convention. Flux uses an area vector normal to the surface, so \Phi=\vec E\cdot\vec A counts the field through the surface, not along the surface.

Figure. Flux Φ = E A cos θ uses the same along-part grammar as work. Face-on (θ = 0) counts the full EA; edge-on (θ = 90°) contributes nothing — the field grazes the surface.

How it works

  1. EquilibriumResolve every force along the same axes and set each component sum to zero.
  2. Field directionA field vector says what direction a test object would be pushed.
  3. FluxUse the surface normal as the area-vector direction.
Electric flux through a flat surface is maximum when the electric field is
  1. Parallel to the surface
  2. Perpendicular to the surface
  3. At any angle because flux is scalar

Area vector is normal to the surface, so \Phi=EA\cos\theta is maximum when field is parallel to the area vector, i.e. perpendicular to the surface.

Notes

  • Scalars have magnitude only; vectors have magnitude and direction and add by components. Resolve \vec A into A_x=A\cos\theta and A_y=A\sin\theta after choosing axes deliberately.
  • Unit vectors carry direction without size: \hat a=\vec A/|\vec A|. In two dimensions, \vec A=A_x\hat i+A_y\hat j and |\vec A|=\sqrt{A_x^2+A_y^2}.
  • The dot product \vec A\cdot\vec B=AB\cos\theta extracts the component along another vector. Work, electric flux and projection questions use this idea directly.
  • The cross product |\vec A\times\vec B|=AB\sin\theta gives a vector perpendicular to the plane by the right-hand rule. Torque \vec\tau=\vec r\times\vec F and angular momentum \vec L=\vec r\times\vec p are physics uses, not abstract geometry tricks.
  • Relative velocity is frame subtraction: \vec v_{A/B}=\vec v_A-\vec v_B. Concurrent-force equilibrium is vector addition to zero, usually by components.

Formulas

  • \vec A=A_x\hat i+A_y\hat j, |\vec A|=\sqrt{A_x^2+A_y^2}
  • A_x=A\cos\theta, A_y=A\sin\theta when \theta is measured from the x-axis
  • \vec A\cdot\vec B=A_xB_x+A_yB_y=AB\cos\theta
  • |\vec A\times\vec B|=AB\sin\theta
  • \vec v_{A/B}=\vec v_A-\vec v_B
  • \sum F_x=0, \sum F_y=0 for equilibrium of concurrent forces

Exam traps & shortcuts

  • Choose axes before resolving; changing axes halfway creates most sign errors.
  • Dot product keeps the parallel part; cross product keeps the perpendicular part.
  • For relative velocity, subtract the observer velocity from the observed velocity, never the other way round.

Reference tables

Physics vector operations
OperationPhysics meaningExamples
Componentprojection on an axismotion, force balance
Dot productparallel partwork, power, flux
Cross productturning/perpendicular effecttorque, angular momentum, magnetic force
Relative velocityframe subtractionrain, boats, projectile separation

Recap

Read only this before a vector-heavy Physics problem.

Resolve
Choose axes first, then components inherit signs.
Dot
Dot product selects the along-part: work, power and flux.
Cross
Cross product selects the perpendicular-effect part: torque, angular momentum and magnetic force.
Relative
Velocity of A seen by B is always v_A-v_B.

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