2Vectors and Scalars

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DIFFERENCES BETWEEN SCALAR AND VECTOR
Property
Scalar
Vector
Definition
Quantities with only magnitude
Quantities with both magnitude and direction
Direction
No direction
Has specific direction
Representation
Represented by a number and unit
Represented by an arrow with length and direction
Change
Changes only with change in magnitude
Changes with change in magnitude or direction or both
Addition
Simple arithmetic addition
Vector addition using parallelogram or triangle law
Examples
Mass, temperature, time, speed
Displacement, velocity, force, acceleration
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TENSORS
Tensors are those quantity whose magnitude differs from direction to direction
▢ Examples:
  1. Pressure
  2. Stress
  3. Modulus of elasticity
  4. Moment of inertia
  5. Coefficient of Viscosity
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SCALARS
▢ Examples:
  • •
    All potentials
  • •
    All fluxes
  • •
    Intensity of energy
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VECTORS
▢ Examples:
  • •
    All flux densities
  • •
    Gradients of all quantities
  • •
    All field strengths/intensities
  • •
    All dipole moments
▢ Special Types:
❖ Polar Vector:
◉ Association: Linear directional effect
◉ Examples:
  • •
    Displacement \(\vec{s}\)
  • •
    Velocity \(\vec{v}\)
  • •
    Acceleration \(\vec{a}\)
  • •
    Momentum \(\vec{p}\)
  • •
    Force \(\vec{F}\)
❖ Axial Vector:
◉ Association: Rotation about an axis
◉ Examples:
  • •
    Angular displacement \(\vec{\theta}\)
  • •
    Angular velocity \(\vec{\omega}\)
  • •
    Angular acceleration \(\vec{\alpha}\)
  • •
    Angular momentum \(\vec{L}\) or \(\vec{J}\)
  • •
    Torque \(\vec{\tau}\)
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TENSOR
▢ Definition: Quantity whose magnitude differs from direction to direction
▢ Examples:
  • •
    Pressure
  • •
    Stress
  • •
    Modulus of elasticity
  • •
    Moment of inertia
  • •
    Coefficient of viscosity
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UNIT VECTOR
▢ Definition: Vector of unit magnitude; direction only
▢ Representation:
  • •
    \(\hat{a}\)
  • •
    \(\hat{a}=\dfrac{\vec{a}}{|\vec{a}|}\)
  • •
    Along \(\hat{i}+\hat{j}+\hat{k}\): \(\dfrac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}\)
▢ Properties:
  • •
    Unitless
  • •
    Dimensionless
  • •
    Possesses only direction
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VECTOR ADDITION
▢ General Resultant:
❖ Notation: \(\vec{R}=\vec{a}+\vec{b};\ a=|\vec{a}|,\ b=|\vec{b}|\)
❖ Magnitude: \(|\vec{R}|=\sqrt{a^2+b^2+2ab\cos\theta}\)
❖ Direction with \(\vec{a}\): \(\alpha=\tan^{-1}\left(\dfrac{b\sin\theta}{a+b\cos\theta}\right)\)
❖ Direction with \(\vec{b}\): \(\beta=\tan^{-1}\left(\dfrac{a\sin\theta}{b+a\cos\theta}\right)\)
▢ Special Cases:
  • •
    \(\theta=0^\circ\Rightarrow |\vec{a}+\vec{b}|=a+b\) → maximum
  • •
    \(\theta=90^\circ\Rightarrow |\vec{a}+\vec{b}|=\sqrt{a^2+b^2}\)
  • •
    \(\theta=180^\circ\Rightarrow |\vec{a}+\vec{b}|=|a-b|\)
  • •
    \(a=b=A\Rightarrow |\vec{a}+\vec{b}|=2A\cos\dfrac{\theta}{2}\)
  • •
    \(a=b=|\vec{a}+\vec{b}|\Rightarrow \theta=120^\circ\)
  • •
    \(\vec{a}+\vec{b}=\vec{c},\ a^2+b^2=c^2\Rightarrow \theta=90^\circ\)
  • •
    \(\vec{a}+\vec{b}=\vec{R}\Rightarrow \vec{a},\vec{b},\vec{R}\) coplanar
▢ Limits: \(|a-b|\le |\vec{a}+\vec{b}|\le a+b\)
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VECTOR SUBTRACTION
▢ General Resultant:
❖ Notation: \(\vec{R}=\vec{a}-\vec{b}\)
❖ Magnitude: \(|\vec{R}|=\sqrt{a^2+b^2-2ab\cos\theta}\)
❖ Direction with \(\vec{a}\): \(\alpha=\tan^{-1}\left(\dfrac{b\sin\theta}{a-b\cos\theta}\right)\)
❖ Direction with \(\vec{b}\): \(\beta=\tan^{-1}\left(\dfrac{a\sin\theta}{b-a\cos\theta}\right)\)
▢ Special Cases:
  • •
    \(\theta=0^\circ\Rightarrow |\vec{a}-\vec{b}|=|a-b|\) → minimum
  • •
    \(\theta=90^\circ\Rightarrow |\vec{a}-\vec{b}|=\sqrt{a^2+b^2}\)
  • •
    \(\theta=180^\circ\Rightarrow |\vec{a}-\vec{b}|=a+b\) → maximum
  • •
    \(a=b=A\Rightarrow |\vec{a}-\vec{b}|=2A\sin\dfrac{\theta}{2}\)
  • •
    \(a=b=A,\ \theta=90^\circ\Rightarrow |\vec{a}-\vec{b}|=\sqrt{2}A\)
  • •
    \(a=b=A,\ \theta=60^\circ\Rightarrow |\vec{a}-\vec{b}|=A\)
  • •
    Vector \(A\) rotated through \(\theta\): change \(=2A\sin\dfrac{\theta}{2}\)
  • •
    \(a=b=|\vec{a}-\vec{b}|\Rightarrow \theta=60^\circ\)
▢ Limits: \(|a-b|\le |\vec{a}-\vec{b}|\le a+b\)
▢ Addition–Subtraction Relations:
  • •
    \(|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|\Rightarrow \theta=90^\circ\)
  • •
    \(\vec{a}+\vec{b}=\vec{a}-\vec{b}\Rightarrow \vec{b}=\vec{0}\)
▢ Properties:
  • •
    Addition commutative: \(\vec{a}+\vec{b}=\vec{b}+\vec{a}\)
  • •
    Subtraction non-commutative: \(\vec{a}-\vec{b}\ne\vec{b}-\vec{a}\)
  • •
    Angle between \(\vec{a}-\vec{b}\) and \(\vec{b}-\vec{a}\): \(180^\circ\)
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VECTOR LAWS
▢ Triangle Law:
❖ Statement: Two vectors represented by two sides of a triangle taken in order → resultant represented by third side from initial point to terminal point
❖ Resultant: \(\vec{R}=\vec{a}+\vec{b}\)
▢ Lami's Theorem:
❖ Condition: Three coplanar concurrent vectors \(\vec{a},\vec{b},\vec{c}\) in equilibrium
❖ Relation: \(\dfrac{a}{\sin\alpha}=\dfrac{b}{\sin\beta}=\dfrac{c}{\sin\gamma}\)
❖ Angle Relation: \(\cos\gamma=\dfrac{c^2-a^2-b^2}{2ab}\)
❖ Cross-product Relation: \(\vec{a}+\vec{b}+\vec{c}=\vec{0}\Rightarrow \vec{a}\times\vec{b}=\vec{b}\times\vec{c}=\vec{c}\times\vec{a}\)
▢ Parallelogram Law:
❖ Statement: Two vectors represented by adjacent sides of a parallelogram → diagonal through common origin gives resultant
❖ Resultant: \(\vec{R}=\vec{a}+\vec{b}\)
▢ Polygon Law:
❖ Statement: Vectors represented by successive polygon sides in the same order → closing side in reverse order gives resultant
❖ Resultant: \(\vec{R}=\vec{a}+\vec{b}+\vec{c}+\cdots\)
❖ Notes:
  • •
    \(\vec{a}+\vec{b}+\vec{c}+\vec{d}=\vec{0}\): vectors may or may not be coplanar
  • •
    Largest magnitude \(\le\) sum of remaining magnitudes → zero resultant possible
  • •
    Example: 6 N, 8 N, 12 N may yield zero resultant
  • •
    Largest magnitude \(>\) sum of remaining magnitudes → zero resultant impossible
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COMPONENTS OF A VECTOR
▢ Rectangular Resolution:
❖ Vector: \(\vec{R}=R_x\hat{i}+R_y\hat{j}\)
❖ Horizontal: \(R_x=R\cos\alpha\)
❖ Vertical: \(R_y=R\sin\alpha\)
❖ Magnitude: \(R=\sqrt{R_x^2+R_y^2}\)
❖ Direction: \(\alpha=\tan^{-1}\left(\dfrac{R_y}{R_x}\right)\)
▢ Equal Components:
  • •
    Plane: 2 equal rectangular components → \(\cos^{-1}\left(\dfrac{1}{\sqrt{2}}\right)=45^\circ\) with each axis
  • •
    Space: 3 equal rectangular components → \(\cos^{-1}\left(\dfrac{1}{\sqrt{3}}\right)\approx54.74^\circ\) with each axis
▢ Properties:
  • •
    A vector can have infinitely many component vectors
  • •
    Plane → 2 rectangular components
  • •
    Space → 3 rectangular components
  • •
    Each rectangular component magnitude \(\le\) vector magnitude
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VECTOR MULTIPLICATION
▢ Scalar or Dot Product:
❖ Definition: \(\vec{a}\cdot\vec{b}=ab\cos\theta\)
❖ Nature: Scalar
❖ Special Cases:
  • •
    \(\theta=0^\circ\Rightarrow \vec{a}\cdot\vec{b}=ab\) → maximum
  • •
    \(\theta=90^\circ\Rightarrow \vec{a}\cdot\vec{b}=0\)
  • •
    \(\theta=180^\circ\Rightarrow \vec{a}\cdot\vec{b}=-ab\) → minimum
  • •
    \(\hat{i}\cdot\hat{i}=\hat{j}\cdot\hat{j}=\hat{k}\cdot\hat{k}=1\)
  • •
    \(\hat{i}\cdot\hat{j}=\hat{j}\cdot\hat{k}=\hat{k}\cdot\hat{i}=0\)
❖ Examples:
  • •
    Work: \(W=\vec{F}\cdot\vec{s}=Fs\cos\theta\)
  • •
    Instantaneous power: \(P=\vec{F}\cdot\vec{v}=Fv\cos\theta\)
  • •
    Magnetic flux: \(\Phi=\vec{B}\cdot\vec{A}=BA\cos\theta\)
❖ Limits: \(-ab\le\vec{a}\cdot\vec{b}\le ab\)
▢ Vector or Cross Product:
❖ Definition: \(\vec{a}\times\vec{b}=ab\sin\theta\,\hat{n}\)
❖ Nature: Vector
❖ Direction: \(\hat{n}\perp\vec{a},\vec{b}\); right-hand screw law/right-hand thumb rule
❖ Orthogonality: Angle between \(\vec{a}+\vec{b}\) and \(\vec{a}\times\vec{b}\): \(90^\circ\)
❖ Special Cases:
  • •
    \(\theta=0^\circ\) or \(180^\circ\Rightarrow \vec{a}\times\vec{b}=\vec{0}\)
  • •
    \(\theta=90^\circ\Rightarrow |\vec{a}\times\vec{b}|=ab\)
  • •
    \(\hat{i}\times\hat{i}=\hat{j}\times\hat{j}=\hat{k}\times\hat{k}=\vec{0}\)
  • •
    \(\hat{i}\times\hat{j}=-\hat{j}\times\hat{i}=\hat{k}\)
  • •
    \(\hat{j}\times\hat{k}=-\hat{k}\times\hat{j}=\hat{i}\)
  • •
    \(\hat{k}\times\hat{i}=-\hat{i}\times\hat{k}=\hat{j}\)
❖ Properties:
  • •
    Non-commutative: \(\vec{a}\times\vec{b}\ne\vec{b}\times\vec{a}\)
  • •
    Anti-commutative: \(\vec{a}\times\vec{b}=-(\vec{b}\times\vec{a})\)
  • •
    \(|\vec{a}\times\vec{b}|=|\vec{b}\times\vec{a}|\)
  • •
    Angle between \(\vec{a}\times\vec{b}\) and \(\vec{b}\times\vec{a}\): \(180^\circ\)
❖ Geometrical Meaning:
◉ Parallelogram Area: \(A=ab\sin\theta=|\vec{a}\times\vec{b}|\)
◉ Minimum: \(\theta=0^\circ\) or \(180^\circ\Rightarrow A=0\)
◉ Maximum: \(\theta=90^\circ\Rightarrow A=ab\)
◉ Using Diagonals: \(A=\dfrac{1}{2}d_1d_2\sin\theta\)
❖ Limits: \(0\le|\vec{a}\times\vec{b}|\le ab\)
❖ Examples:
◉ Axial Vectors:
  • •
    Angular momentum: \(\vec{L}=\vec{r}\times\vec{p}\)
  • •
    Torque: \(\vec{\tau}=\vec{r}\times\vec{F}\)
◉ Linear Vectors:
  • •
    Linear displacement: \(\vec{s}=\vec{\theta}\times\vec{r}\)
  • •
    Linear velocity: \(\vec{v}=\vec{\omega}\times\vec{r}\)
  • •
    Tangential acceleration: \(\vec{a}_t=\vec{\alpha}\times\vec{r}\)
  • •
    Centripetal acceleration: \(\vec{a}_c=\vec{\omega}\times\vec{v}\)
▢ Dot–Cross Relations:
  • •
    \(\vec{a}\cdot\vec{b}=|\vec{a}\times\vec{b}|\Rightarrow \theta=45^\circ\)
  • •
    Coplanarity: \(\vec{a}\cdot(\vec{b}\times\vec{c})=0\)
  • •
    \(\vec{a}\cdot\vec{b}=0,\ \vec{a}\times\vec{c}=\vec{0}\Rightarrow \angle(\vec{b},\vec{c})=90^\circ\)
▢ Angle Between Two Vectors:
❖ Sine: \(\sin\theta=\dfrac{|\vec{A}\times\vec{B}|}{AB}\)
❖ Cosine: \(\cos\theta=\dfrac{\vec{A}\cdot\vec{B}}{AB}\)
❖ Tangent: \(\tan\theta=\dfrac{|\vec{A}\times\vec{B}|}{\vec{A}\cdot\vec{B}}\)
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PROJECTION
▢ Scalar Projection of \(\vec{b}\) on \(\vec{a}\):
  • •
    \(OM=b\cos\theta\)
  • •
    \(OM=\dfrac{\vec{a}\cdot\vec{b}}{a}\)
  • •
    \(OM=\vec{b}\cdot\hat{a}\)
▢ Vector Component of \(\vec{b}\) along \(\vec{a}\):
  • •
    \(\vec{OM}=(b\cos\theta)\hat{a}\)
  • •
    \(\vec{OM}=(\vec{b}\cdot\hat{a})\hat{a}\)
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MINIMUM VECTORS FOR ZERO RESULTANT

Table 1: Minimum number

Configuration
Minimum vectors
Collinear vectors
2
Collinear vectors unequal in magnitude
3
Coplanar vectors
3
Non-coplanar vectors
4
Q1.
The resultant of two force 3P and 2P is R. If the first force is doubled then the resultant is also doubled. The angle between the forces is:
📅CEE Nursing 2024
Q2.
Which of the following is a vector quantity?
📅CEE MBBS 2024B
Q3.
Which of the following is a scalar quantity?
📅CEE MBBS 2024a
Q4.
If P·Q = 0 and P×R = 0, what is the angle between vectors Q and R?
📅CEE BDS 2024
Q5.
A person who can swim at 5 km/hr in still water cross a river 1 km wide flowing at 3 km/hr in shortest time. How far will he reach at another bank?
📅CEE BDS 2021
Q6.
A person moves with velocity components \(10\ \mathrm{km\,h^{-1}}\) east and \(20\ \mathrm{km\,h^{-1}}\) north. Find the magnitude of the resultant velocity.
📅BP 2014•BP 2016
Q7.
For vectors of magnitudes \(A\) and \(B\), what is the maximum possible value of \(|\vec A-\vec B|\)?
📅BP 2013
Q8.
Given \(\vec A=5\hat i+7\hat j-3\hat k\) and \(\vec B=2\hat i+2\hat j-a\hat k\), find \(a\) if the vectors are perpendicular.
📅BP 2009
Q9.
A vector of magnitude \(l\) rotates through an angle \(\theta\) about its tail. Find the displacement of its head.
📅BP 2009
Q10.
Two forces of magnitudes \(3p\) and \(2p\) have resultant \(R\). If the first force is doubled, the resultant becomes \(2R\). Find the angle between the forces.
📅IOM 2009•KU 2014•KU 2013
Q11.
Which of the following is a vector?
📅IOM 2012
Q12.
Two velocities \(\vec v_1\) and \(\vec v_2\) make an angle \(\theta\). If \(\vec v_1\) lies along the positive x-axis, what is the x-component of \(\vec v_1-\vec v_2\)?
📅MOE 2013
Q13.
Body A moves north at \(3\ \mathrm{km/s}\) while B moves east at \(4\ \mathrm{km/s}\). What is the velocity of A relative to B?
📅MOE 2011
Q14.
Which operation leaves a free vector unchanged?
📅KU 2010
Q15.
A body moves \(30\ \mathrm m\) north, \(20\ \mathrm m\) east and \(30\sqrt2\ \mathrm m\) southwest. Find its net displacement.
📅IE 2011
Q16.
The x-component of a vector making \(30^\circ\) with the positive x-axis is 3 units. Find its y-component.
📅Bangladesh 2009
Q17.
Which of the following is a scalar quantity?
📅KU 2009
Q18.
Two forces each have magnitude \(F\). Their resultant also has magnitude \(F\). Find the angle between them.
📅IOM 2008
Q19.
Two vectors of magnitudes 5 and 12 units are perpendicular. Their resultant has magnitude 13 units. What is the angle between the resultant and the vector of magnitude 12?
📅IOM 2005
Q20.
The resultant of two forces \(\vec F_1\) and \(\vec F_2\) is perpendicular to \(\vec F_1\). If its magnitude is also \(F_1\), find \(F_2\).
📅IOM 2002•IOM 2001
Q21.
Magnetic moment is
Q22.
The sum and the difference of two nonzero vectors have equal magnitudes. What is the angle between the vectors?
📅IOM 1997
Q23.
Which of the following is not a vector quantity?
📅MOE 2063
Q24.
If \(|\vec A|=6,\ |\vec B|=8,\ |\vec A+\vec B|=10\), what is the angle between A and B?
📅MOE 2056
Q25.
Forces of 1 N, 3 N and 2 N act at angles \(0^\circ,90^\circ,120^\circ\) with the positive x-axis. In which direction is the resultant?
📅MOE 2054
Q26.
For vectors \(\vec A\) and \(\vec B\), what condition makes \(\vec A+\vec B=\vec A-\vec B\)?
📅BPKIHS 2005
Q27.
Which of the following can be the magnitude of the resultant of forces 8 N and 6 N?
Q28.
For two forces \(\vec P\) and \(\vec Q\), the resultant \(\vec R\) is perpendicular to \(\vec P\) and \(R=P\). Find \(Q\).
Q29.
The sum of two unit vectors is also a unit vector. Find the magnitude of their difference.
Q30.
Two vectors have magnitudes 4 and 3, and their dot product is 6. Find the angle between them.
Q31.
The sum and difference of two vectors are perpendicular. What follows?
📅BP 2017
Q32.
The dot product of two nonzero vectors is \(\sqrt3\) times the magnitude of their cross product. Find the angle between them.
Q33.
If two nonzero vectors \(\vec a\) and \(\vec b\) are equal, which statement is true?
Q34.
What is \(\vec A\cdot(\vec B\times\vec A)\) for any vectors \(\vec A,\vec B\)?
Q35.
Which magnitude cannot be the resultant of vectors of magnitudes 5 and 10?
Q36.
Find the scalar component of \(\vec A=2\hat i+3\hat j\) along \(\vec B=\hat i+\hat j\).
Q37.
If \(\vec a=\vec b+\vec c\), with \(a=5,b=4,c=3\) and \(\vec b\perp\vec c\), find the angle between \(\vec a\) and \(\vec c\).
Q38.
Which operation is not generally defined for vectors in elementary vector algebra?
Q39.
If \(|\vec a\cdot\vec b|=|\vec a\times\vec b|\) and the angle between the vectors is acute, find \(|\vec a+\vec b|\).
Q40.
Which three force magnitudes can never produce zero net force, regardless of their directions?
📅IOM
Q41.
Find a unit vector along \(\hat i+\hat j\).
Q42.
The diagonals of a parallelogram are \(\vec d_1=(2\hat i+2\hat j)\ \mathrm{cm}\) and \(\vec d_2=(2\hat i-2\hat j)\ \mathrm{cm}\). Find its area.
Q43.
Find the scalar projection of \(\hat i+2\hat j+3\hat k\) on \(\hat i+\hat j+\hat k\).
Q44.
A hall measures 12 m × 4 m × 3 m. What is the displacement between two opposite corners?
Q45.
Forces \(\vec F_1=(3\hat i+4\hat j)\ \mathrm N\) and \(\vec F_2=(4\hat i+3\hat j)\ \mathrm N\) act together. Find the magnitude of their resultant.
Q46.
Two nonzero vectors \(\vec P\) and \(\vec Q\) make an angle \(120^\circ\). Compare \(|\vec P+\vec Q|\) and \(|\vec P-\vec Q|\).
📅KU 2015
Q47.
Two vectors have equal magnitude \(V\). Which value cannot be the magnitude of their resultant?
📅KU 2016