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 \(\vec{\omega}\)
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    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
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    \(\theta=90^\circ\Rightarrow |\vec{a}+\vec{b}|=\sqrt{a^2+b^2}\)
  • •
    \(\theta=180^\circ\Rightarrow |\vec{a}+\vec{b}|=|a-b|\)
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    \(a=b=A\Rightarrow |\vec{a}+\vec{b}|=2A\cos\dfrac{\theta}{2}\)
  • •
    \(a=b=|\vec{a}+\vec{b}|\Rightarrow \theta=120^\circ\)
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    \(\vec{a}+\vec{b}=\vec{c},\ a^2+b^2=c^2\Rightarrow \theta=90^\circ\)
  • •
    \(\vec{a}+\vec{b}=\vec{R}\Rightarrow \vec{a},\vec{b},\vec{R}\) 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.
A man goes 10 km/hr east and 20 km/hr north. Find the relative velocity.
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Q2.
What will be the maximum magnitude of (A - B)?
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Q3.
Two vectors A = 5i + 7j - 3k and B = 2i + 2j - ak are perpendicular to each other, then value of 'a' is
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Q4.
A vector of length l is turned through the angle θ about its tail. What is the change in the position vector of its head?
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Q5.
The resultant of two forces 3p and 2p is R. If the first force is doubled then the resultant is also doubled. The angle between the forces is
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Q6.
Which of the following is a vector?
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Q7.
Two bodies are moving with velocities V₁ and V₂ respectively. V₁ is along X-axis and V₂ moving in the first quadrant, makes an angle θ with V₁. The relative velocity of V₁ with respect to x-component of V₂ will be
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Q8.
A body A moving north with 3 km/s and B with 4 km/s east. What is the relative velocity of A with respect to B?
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Q9.
A vector remains unchanged
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Q10.
A body moves 30 m due north, 20 m due east and 30√2 m due southwest. The total displacement covered by the body from its initial position is
📅IE 2011
Q11.
The x-component of a vector making an angle of 30° with horizontal is 3. Its y-component is
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Q12.
Which of the following is a scalar quantity?
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Q13.
Two forces of magnitude F have resultant of the same magnitude F. The angle between the two forces is
📅IOM 2008
Q14.
Three vectors are arranged to form a right-angled triangle of sides 5, 12 and 13 units. The sum of the two vectors is equal to the third. The angle between those of magnitudes 12 and 13 will be
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Q15.
Resultant of two forces F₁ and F₂ is R and the resultant is at right angle to the force F₁. Then the force F₂ is equal to
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Q16.
Magnetic moment is
Q17.
Two vectors have a sum A and a difference B. If A = B then, the angle between the two vectors is
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Q18.
Which of the following is not a vector quantity?
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Q19.
If A, B, C have magnitudes 6, 8 & 10 respectively, and A + B = C, the angle between A & B is
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Q20.
Three forces of magnitudes 1N, 3N and 2N are acting at angles of 0°, 90° and 120° with +ve X-axis respectively, then the resultant will act along the:
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Q21.
The condition for A + B = A - B is that:
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Q22.
The resultant of two forces 8N and 6N is
Q23.
The resultant of two forces P and Q is perpendicular to P and is equal to P. Then the magnitude of another force Q is
Q24.
The sum of two unit vectors is a unit vector. Then their difference will be
Q25.
The dot product of two vectors is 6 and their magnitudes are 4 and 3. Then angle between these vectors will be
Q26.
The vector sum and vector difference of two vectors are at right angle. Then these vectors
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Q27.
The dot product of vectors is √3 times the magnitude of their cross product. Then angle between these vectors will be
Q28.
If a = b then
Q29.
The angle between A and B is θ. The value of A.BxA is
Q30.
Which of the following can't be resultant of the vectors of magnitude 5 and 10?
Q31.
What is the component of A = 2i + 3j along B = i +j?
Q32.
If a = b + c and |a|=5, |b|=4, |c|=3, the angle between a and c is
Q33.
Which of the following is not defined in vectors?
Q34.
If |a.b|= |axb|, then |a+b|
Q35.
Which set of forces acting on a body never produces zero acceleration?
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Q36.
The unit vector along i + j is
Q37.
Two diagonals of a parallelogram are (2i + 2j) and (2i - 2j) cm. Then area of the parallelogram will be
Q38.
What is the projection of i + 2j + 3k on i + j + k?
Q39.
The length, breadth and height of a hall are 12m, 4m and 3m. What is the displacement of a fly which flies from one corner to another corner of the hall?
Q40.
The forces F₁ = (3i + 4j) N and F₂ = (4i + 3j) N are acting on a body. The resultant force on the body is
Q41.
Let the angle between two non-zero vectors P and Q be 120° and its resultant be R. Then
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Q42.
If the two vectors V and V₁ have the same magnitude, then which of the following is not true for the sum of their magnitude?
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