Vectors and equilibrium: Class 11 Physics

Key points

A scalar is a quantity that has magnitude only, such as mass, time, speed, energy and temperature. A vector has both magnitude and direction, such as displacement, velocity, force and acceleration. Vectors add by special rules, not by ordinary arithmetic. In print a vector is written in bold or with an arrow, for example A or A⃗.

A vector is drawn as a straight line with an arrowhead. The length of the line (to a chosen scale) shows the magnitude and the arrow shows the direction. Two vectors are equal if they have the same magnitude and the same direction. A negative vector −A has the same magnitude as A but the opposite direction.

Addition of vectors by the head to tail rule:

• Draw the first vector A to scale.
• Draw the second vector B so that its tail starts at the head of A.
• The resultant R = A + B is drawn from the tail of A to the head of B.
• The order does not matter, so A + B = B + A. For many vectors, join them head to tail one after another and draw the resultant from the first tail to the last head.

Subtraction is addition of the negative vector: A − B = A + (−B). For two vectors of magnitudes A and B with angle θ between them, the magnitude of the resultant is R = √(A² + B² + 2AB cos θ). Its direction is found from tan α = B sin θ / (A + B cos θ), where α is measured from A. When θ = 0° the resultant is largest (A + B). When θ = 180° it is smallest (A − B).

Resolution is the opposite of addition. A vector F that makes an angle θ with the x axis can be split into two perpendicular components: Fx = F cos θ along x and Fy = F sin θ along y. Then F = √(Fx² + Fy²) and tan θ = Fy/Fx. Resolving forces into perpendicular parts is the standard method for solving problems on inclined planes and projectiles.

A unit vector has magnitude 1 and shows direction only. The unit vectors along the x, y and z axes are i, j and k. Any vector can be written as A = Ax i + Ay j + Az k, and its magnitude is |A| = √(Ax² + Ay² + Az²). To add vectors, add the matching components. A unit vector in the direction of A is A/|A|.

The scalar (dot) product:

• A·B = AB cos θ, where θ is the angle between the vectors. The result is a scalar.
• It is commutative: A·B = B·A.
• i·i = j·j = k·k = 1 and i·j = j·k = k·i = 0.
• In components, A·B = AxBx + AyBy + AzBz.
• Two non-zero vectors are perpendicular if A·B = 0. Work, W = F·d, is a dot product.

The vector (cross) product:

• A × B = AB sin θ n, where n is a unit vector perpendicular to both A and B. Its direction is given by the right hand rule: curl the fingers from A to B and the thumb points along A × B.
• It is not commutative: A × B = −(B × A).
• i × j = k, j × k = i, k × i = j, and the product of parallel vectors is zero.
• The magnitude AB sin θ equals the area of the parallelogram formed by A and B.

The torque (moment of a force) measures the turning effect of a force about a point or axis. τ = r × F, so its magnitude is τ = rF sin θ = F × d, where d = r sin θ is the perpendicular distance from the axis to the line of action of the force. Its SI unit is the newton metre (N m). A longer spanner gives a bigger torque for the same force. Torque is a vector, and anticlockwise torques are usually taken as positive.

Conditions for equilibrium:

• A body is in equilibrium when it has no linear acceleration and no angular acceleration.
• First condition: the vector sum of all forces is zero, ΣF = 0, so ΣFx = 0 and ΣFy = 0.
• Second condition: the sum of all torques about any point is zero, Στ = 0. Clockwise moments equal anticlockwise moments.
• In static equilibrium the body is at rest. In dynamic equilibrium it moves with constant velocity.

A couple is a pair of equal and opposite parallel forces whose lines of action do not coincide. The resultant force of a couple is zero, but it produces a turning effect. Its moment is the force times the perpendicular distance between the two lines of action: τ = F × d. Turning a tap or a steering wheel with two hands applies a couple. To solve a problem on equilibrium, draw the free body diagram, choose a convenient pivot (often where an unknown force acts, so that force has zero moment), and write both conditions.

Study tip: Always begin with a neat diagram and mark every force. Show the resolution into components before using them. For equilibrium problems, write both conditions separately and take moments about the point where an unknown force acts. State units with every answer.

Definitions and flashcards

Say the meaning first, then open the card to check.

Formula sheet

Print this page

Short questions with model answers

These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.

Numericals with worked steps

Common mistakes

• Adding magnitudes of vectors directly. 3 N and 4 N do not give 7 N unless they act in the same direction.
• Using cos θ for a component when θ is measured from the other axis. If the angle is from the vertical, the vertical component is F cos θ.
• Mixing up the dot and cross products. The dot product is a scalar with cos θ, the cross product is a vector with sin θ.
• Measuring the arm of a torque along the force direction instead of the perpendicular distance from the axis to the line of action.
• Writing moments about a point but forgetting one of the forces, such as the weight of the beam acting at its centre.

Practise this chapter

Practice questions and timed test Printable worksheet