Vectors and Equilibrium notes

MDCAT Physics

This chapter explains scalar and vector quantities, methods of adding and resolving vectors, rectangular components, and scalar and vector products. It also describes equilibrium, torque, position vectors, unit vectors, and the conditions needed for a body to remain balanced.

Scalars and Vectors

A scalar quantity has magnitude only. A vector quantity has both magnitude and direction. Scalars are added by ordinary algebra, while vectors must be added by considering their directions.

A vector is represented by a directed line segment. The length of the line represents its magnitude, and the arrowhead shows its direction. A vector may be written with a bold letter or an arrow over its symbol, such as A or vector A.

Examples of scalars include mass, time, temperature, distance, speed, work, power, energy, electric charge, and electrostatic potential.
Examples of vectors include displacement, velocity, acceleration, force, momentum, angular momentum, and torque.
The magnitude of vector A is written as |A| or A.
A vector remains unchanged if it is shifted parallel to itself without changing its magnitude and direction.
Multiplying a vector by a positive number changes its magnitude but not its direction.
Multiplying a vector by a negative number reverses its direction.
A zero vector has zero magnitude and no definite direction.
Work and power are both scalar quantities, whereas force and displacement are vector quantities.

Representation and Unit Vectors

A unit vector is a vector having magnitude equal to one. It is used to specify direction. Along the mutually perpendicular x, y, and z axes, the unit vectors are i, j, and k respectively.

A vector can be expressed in terms of its rectangular components as A = Ax i + Ay j + Az k. Here Ax, Ay, and Az are the components along the x, y, and z axes.

i is the unit vector along the positive x-axis.
j is the unit vector along the positive y-axis.
k is the unit vector along the positive z-axis.
The magnitudes of i, j, and k are all equal to 1.
i, j, and k are mutually perpendicular.
i . j = 0, j . k = 0, and k . i = 0.
i . i = 1, j . j = 1, and k . k = 1.
For B = i - 2j + 2k, |B| = √(1 + 4 + 4) = 3, so the unit vector along B is B/3.
A position vector r gives the position of a point relative to a chosen origin.

Addition of Vectors and Resultant

The vector obtained by adding two or more vectors is called their resultant. Vector addition must take both magnitude and direction into account. The order of addition does not affect the resultant, so vector addition is commutative.

The triangle law states that if two vectors are represented by two sides of a triangle taken in order, their resultant is represented by the third side drawn from the initial point to the final point. The parallelogram law states that if two vectors acting at a point are represented by the adjacent sides of a parallelogram, the diagonal through that point represents their resultant.

For two vectors A and B making angle θ, the magnitude of the resultant is R = √(A² + B² + 2AB cos θ).
The direction α of the resultant relative to A is given by tan α = B sin θ/(A + B cos θ).
If two vectors are parallel and in the same direction, R = A + B.
If two vectors are anti-parallel, R = |A - B|.
The maximum resultant of vectors A and B is A + B.
The minimum resultant of vectors A and B is |A - B|.
For vectors of 10 N and 15 N, the resultant must lie from 5 N to 25 N. Therefore, 30 N cannot be their resultant.
A vector can be reconstructed from its components by vector addition or composition of vectors.

Resolution and Rectangular Components

Resolution of a vector is the process of replacing one vector by two or more component vectors. For a vector in a plane, it is usually resolved into perpendicular x and y components. These components together have the same effect as the original vector.

If vector A makes angle θ with the positive x-axis, its rectangular components are Ax = A cos θ and Ay = A sin θ. The signs of the components depend on the quadrant in which the vector lies.

The process of replacing one vector by two or more parts is called resolution of vectors.
The x-component of A is Ax = A cos θ when θ is measured from the x-axis.
The y-component of A is Ay = A sin θ when θ is measured from the x-axis.
The magnitude of a vector from its rectangular components is A = √(Ax² + Ay²).
The angle made by the vector with the x-axis is given by tan θ = Ay/Ax, with attention to the signs of Ax and Ay.
A vector may have two rectangular components in a plane and three rectangular components in space.
The number of possible components of a vector is not limited to two or three. Components can be taken along any set of suitable directions, so the maximum number may be infinite.
For velocity components 3 m s-1 and 4 m s-1, the speed is √(3² + 4²) = 5 m s-1.
If Ax = √3 and Ay = 1, then tan θ = 1/√3, so θ = 30 degrees when the vector lies in the first quadrant.

Scalar Product or Dot Product

The scalar product of two vectors is an operation that gives a scalar quantity. It is also called the dot product and is represented by a dot between the vectors.

For vectors A and B, A . B = AB cos θ, where θ is the smaller angle between them. In component form, A . B = AxBx + AyBy + AzBz.

The dot product of two vectors is a scalar quantity.
The scalar product of L and M is L . M = LM cos θ.
If two vectors are perpendicular, θ = 90 degrees and A . B = 0.
If two vectors are parallel in the same direction, A . B = AB.
If two vectors are anti-parallel, θ = 180 degrees and A . B = -AB.
The dot product is negative when the angle between the vectors is greater than 90 degrees. For exactly anti-parallel vectors, it is negative.
The dot product is commutative: A . B = B . A.
The work done by a constant force is W = F . s = Fs cos θ.
Electrostatic potential is a scalar quantity.
Using unit vectors, k . j = 0 because k and j are perpendicular.

Vector Product or Cross Product

The vector product of two vectors is also called the cross product. It gives a vector perpendicular to the plane containing the two vectors. It is represented by a cross sign.

For vectors A and B, A × B = AB sin θ n, where n is a unit vector perpendicular to the plane of A and B. The direction is found by the right-hand rule.

The magnitude of the cross product is |A × B| = AB sin θ.
The cross product of parallel or anti-parallel vectors is zero because sin 0 degrees and sin 180 degrees are both zero.
The cross product of perpendicular vectors has maximum magnitude, AB.
A × B is perpendicular to both A and B.
The cross product is anti-commutative: A × B = -(B × A).
i × j = k, j × k = i, and k × i = j.
j × i = -k, k × j = -i, and i × k = -j.
Angular momentum is an example of a vector product: L = r × p.
Torque is a vector product: τ = r × F.
The direction of A × B is found by curling the fingers of the right hand from A toward B. The thumb gives the direction of the product.

Torque, Moment Arm, and Equilibrium

Torque is the turning effect of a force about a fixed point or axis. It depends on the force and the perpendicular distance between the axis and the line of action of the force. This perpendicular distance is called the moment arm.

A body is in equilibrium when it has no linear acceleration and no angular acceleration. Translational equilibrium requires the resultant force to be zero. Rotational equilibrium requires the resultant torque to be zero.

Torque is given by τ = r × F, and its magnitude is τ = rF sin θ.
The moment arm is the perpendicular distance from the axis of rotation to the line of action of the force.
If the moment arm is zero, torque is zero.
A force produces maximum torque when it acts at 90 degrees to the position vector.
A force acting through the axis of rotation produces no torque.
For translational equilibrium, ΣF = 0.
For rotational equilibrium, Στ = 0.
A body in complete equilibrium satisfies both ΣF = 0 and Στ = 0.
Clockwise and anticlockwise torques are taken with opposite signs when solving equilibrium problems.

Useful Relations and Direction Rules

Vector equations can be simplified by using components and unit-vector relations. The signs of components and products must be retained. A result of zero may occur because vectors are perpendicular in a dot product, parallel in a cross product, or equal and opposite in vector addition.

Mathematical operators must also be applied according to their usual order. Division and multiplication are performed before addition and subtraction, while an equality sign separates the two sides of an equation.

The expression 18 ÷ 6 × 9 = 27 is correct because 18 ÷ 6 × 9 = 27.
For a vector A = Ax i + Ay j + Az k, its magnitude is √(Ax² + Ay² + Az²).
The unit vector in the direction of A is A/|A|, provided A is not the zero vector.
If A × B points along the positive z-axis, A and B may lie in the xy-plane with the direction from A toward B producing positive z by the right-hand rule.
A dot product gives a scalar, while a cross product gives a vector.
A zero dot product does not always mean both vectors are zero. It can indicate perpendicular non-zero vectors.
A zero cross product can result from parallel, anti-parallel, or zero vectors.
The resultant of two vectors is found by addition, while the components of one vector are obtained by resolution.

Key terms

Scalar: A physical quantity described completely by magnitude only.
Vector: A physical quantity having both magnitude and direction.
Magnitude: The numerical size or length of a vector.
Unit vector: A vector of magnitude one used to represent direction.
Resultant: The single vector equal to the vector sum of two or more vectors.
Triangle law: A method of vector addition in which vectors are placed head to tail and the resultant joins the first tail to the last head.
Parallelogram law: A method in which two vectors from one point form adjacent sides of a parallelogram and the diagonal gives the resultant.
Resolution of vectors: The process of replacing one vector by two or more component vectors.
Rectangular components: The mutually perpendicular components of a vector, usually along the x, y, and z axes.
Dot product: An operation A . B = AB cos θ that produces a scalar.
Cross product: An operation A × B = AB sin θ n that produces a vector perpendicular to both vectors.
Position vector: A vector drawn from the origin to the position of a point.
Torque: The turning effect of a force about an axis, given by τ = r × F.
Moment arm: The perpendicular distance from the axis to the line of action of a force.
Equilibrium: The condition in which the resultant force and resultant torque are both zero.
Anti-parallel vectors: Vectors along the same line but in opposite directions.

Test yourself on Vectors and Equilibrium

Free Vectors and Equilibrium MCQs with an explanation on every answer. No account needed.

Practise Vectors and Equilibrium MCQs

One page notes

The whole chapter on one page, for the last days before the exam.

Vectors and Equilibrium One Page Notes

Scalars and Vectors

Scalar has magnitude and proper units only. Examples: mass, time, temperature, electric current.
Vector has magnitude, proper units and direction. Examples: displacement, velocity, acceleration, force.
Unit vector has magnitude one and specifies direction.
Position vector extends from origin to a point: r = xi + yj + zk.
Equal opposite vectors have equal magnitudes and directions differing by 180 degrees.

Vector Addition

Resultant vector is the vector sum of two or more vectors.
Two equal opposite vectors have zero resultant magnitude.
If |P + Q| = |P − Q|, then P and Q are perpendicular.
Resultant is zero when magnitudes are equal and the angle is 180 degrees.
Equilibrium requires resultant force zero. The body may be at rest or in uniform motion.

Components and Quadrants

For vector A at angle theta with x-axis: Ax = A cos theta, Ay = A sin theta.
A = Axi + Ayj; magnitude A = square root of (Ax² + Ay²).
Unit vector along 2i − 4j is (i − 2j) / square root of 5.
cos theta i + sin theta j is a unit vector at angle theta to positive x-axis.
Both x and y components negative indicate the third quadrant.
Point (−6, −8) lies in the third quadrant.

Dot Product

A · B = AB cos theta. It is a scalar quantity.
Dot product is commutative and distributive: A · B = B · A; A · (B + C) = A · B + A · C.
For nonzero vectors, A · B = 0 means the vectors are perpendicular.
Dot product is negative for an obtuse angle, including anti-parallel vectors.
For S = 4, R = 6 and S · R = 13.5, theta is approximately 55.77 degrees.

Cross Product

A × B = AB sin theta n-hat. It is a vector perpendicular to both vectors.
A × B = 0 if either vector is null or the vectors are parallel.
For perpendicular vectors, |A × B| = AB.
Two vectors in the xy-plane have cross product along the perpendicular z-axis.
If a × b points along positive z-axis, a and b lie in the xy-plane with suitable order.
Cross product of force and moment arm gives torque.

Torque and Equilibrium

Torque is the product of force and perpendicular distance from axis to its line of action.
Torque magnitude: tau = F r perpendicular = rF sin theta.
Equilibrium requires zero resultant force and zero resultant torque.
A body in equilibrium can be stationary or move with uniform velocity.
Electric current is a scalar quantity; acceleration is a vector quantity.

Must remember

Vector needs magnitude, units and direction. Scalar needs magnitude and units only.
|P + Q| = |P − Q| means the angle between P and Q is 90 degrees.
A × B = 0 means a null vector exists or the vectors are parallel.
A · B is negative when the angle is obtuse, especially for anti-parallel vectors.
For perpendicular vectors, dot product is zero and cross-product magnitude is AB.
A vector's x-component is A cos theta, not A sin theta.
Negative x and y components place a vector in quadrant III.
Torque uses perpendicular distance, not the complete distance from the axis.
Equilibrium permits rest or uniform motion.

Vectors and Equilibrium mnemonics

Graphical addition and possible resultant

Head to tail, first to last. Difference to sum, the result can pass.

Head to tail: Place the tail of the next vector at the head of the previous vector.
First to last: Draw the resultant from the tail of the first vector to the head of the last vector.
Difference to sum: For magnitudes A and B, the resultant lies from |A - B| to A + B.
The result can pass: Forces of 3 N and 6 N can produce an 8 N resultant.
Order: Do not alter the order of vectors when using graphical addition.

Use the triangle inequality to check whether a proposed resultant is possible.

Rectangular components of a vector

Cos hugs X, sin climbs Y. Components meet at ninety.

Cos hugs X: Ax = A cos theta.
Sin climbs Y: Ay = A sin theta.
Components meet at ninety: The rectangular components along the x-axis and y-axis are perpendicular.
X: The x-axis is the horizontal axis.
Example: For 8 N at 30 degrees, Fx = 8 cos 30 degrees = 4√3 N and Fy = 8 sin 30 degrees = 4 N.

Measure theta from the positive x-axis before assigning cosine to x and sine to y.

Scalar product and unit vectors

Dot uses COS. Same is one, ninety is zero, opposite is minus.

Dot uses COS: A dot B = AB cos theta, and the answer is a scalar.
Same is one: For unit vectors, i dot i = j dot j = k dot k = 1.
Ninety is zero: i dot j = j dot k = k dot i = 0, and perpendicular vectors have zero scalar product.
Opposite is minus: Anti-parallel vectors have a negative scalar product.
Parallel unit vectors: If the scalar product of two unit vectors is 1, they are parallel.

The sign of A dot B follows cos theta: positive for parallel, zero for perpendicular and negative for anti-parallel vectors.

Vector product and its direction

Cross uses SIN. Right hand wins, parallel dies, normal flies.

Cross uses SIN: A cross B = AB sin theta n.
Right hand wins: Use the right-hand rule to find the direction of the cross product.
Parallel dies: The cross product of parallel vectors is zero.
Normal flies: The cross product is perpendicular to both original vectors.
Positive z: If a cross b points along the positive z-axis, a and b lie in the xy-plane.
Axial: Angular momentum is an axial vector and is an example of a vector product.

Do not confuse the cross product with the dot product. Cross product gives a vector, while dot product gives a scalar.

Finding vectors from their sum and difference

Add for A, subtract for B. Divide both by two.

Add for A: If S = A + B and D = A - B, then A = (S + D) / 2.
Subtract for B: B = (S - D) / 2.
Divide both by two: The sum or difference must be halved after combining the vectors.
Example: If A + B = 7i + 7k and A - B = -i + k, then A = 3i + 4k.
Magnitude: The magnitude of A = √(3² + 4²) = 5.

Keep the vector components together while adding or subtracting, then calculate the magnitude using the Cartesian formula.

Physics shortcuts

Comparing distance and displacement

Distance equals the magnitude of displacement only when the particle travels along a straight path without reversing direction.

Check whether the path is straight and one-directional.
If yes, distance = |displacement|.
Example: A particle moves 5 m east in a straight line. Distance = 5 m and displacement magnitude = 5 m.

This shortcut does not apply to a curved path or to motion involving a change of direction.

Projectile range and components

For a projectile launched and landing at the same level, use R = u² sin 2θ/g. Resolve the initial velocity into horizontal and vertical components when needed.

Write ux = u cos θ and uy = u sin θ.
For the same launch and landing level, R = u² sin 2θ/g.
Example: u = 20 m/s, θ = 30°, g = 10 m/s². R = 400 sin 60°/10 = 34.6 m.

The range formula does not apply directly when the projectile lands at a different height.

15 more Physics shortcuts are in the MDCAT pack. Already have it? See all shortcuts