QUIZ

Vector Addition

Definition / Explanation / Theory

If

\[\mathbf{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}, \quad \mathbf{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k}\]

then the vector addition represents the combined effect of two or more vectors acting simultaneously.
It gives a resultant vector 𝐑 which has both magnitude and direction.
Geometrically, vector addition follows the parallelogram law or triangle law, depending on how vectors are arranged.

Important Formulas

\[\mathbf{R} = \mathbf{A} + \mathbf{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j} + (A_z + B_z)\hat{k}\] \[|\mathbf{R}| = \sqrt{(A_x + B_x)^2 + (A_y + B_y)^2 + (A_z + B_z)^2}\] \[R = \sqrt{A^2 + B^2 + 2AB\cos\theta}, \quad \tan\alpha = \frac{B\sin\theta}{A + B\cos\theta}\]

where α is the angle of the resultant with vector A.

Important Points

  • Commutative Law: \(\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}\)
  • Associative Law: \((\mathbf{A} + \mathbf{B}) + \mathbf{C} = \mathbf{A} + (\mathbf{B} + \mathbf{C})\)
  • Distributive Law: \(a(\mathbf{A} + \mathbf{B}) = a\mathbf{A} + a\mathbf{B}\)
  • The zero vector acts as the additive identity: \(\mathbf{A} + \mathbf{0} = \mathbf{A}\)
  • The negative of a vector reverses its direction: \(\mathbf{A} + (-\mathbf{A}) = \mathbf{0}\)
  • Vector addition is independent of coordinate rotation (translation invariant).
  • Graphically, if two vectors form adjacent sides of a parallelogram, the diagonal represents their sum.

Scalar (Dot) Product

Definition / Explanation / Theory

The scalar product (or dot product) of two vectors 𝐀 and 𝐁 is defined as

\[\mathbf{A} \cdot \mathbf{B} = AB \cos\theta\]

where θ is the angle between them.
It gives a scalar quantity, representing the magnitude of one vector in the direction of another.

Important Formulas

  • In Cartesian components:

    \[\mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z\]

  • Work done by a force:

    \[W = \mathbf{F} \cdot \mathbf{d} = Fd\cos\theta\]

  • Projection of 𝐀 on 𝐁:

    \[A_{\parallel B} = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|}\]

Important Points

  • The result of the dot product is scalar, not a vector.
  • Measures how much of one vector acts in the direction of another.
  • Commutative: \(\mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A}\)
  • Distributive: \(\mathbf{A} \cdot (\mathbf{B} + \mathbf{C}) = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \cdot \mathbf{C}\)
  • If \(\mathbf{A} \cdot \mathbf{B} = 0\), then vectors 𝐀 and 𝐁 are orthogonal (perpendicular).
  • Used in calculating work, power, projection, and angle between two vectors.
  • Angle between vectors:
    \(\cos\theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}||\mathbf{B}|}\)

Vector (Cross) Product

Definition / Explanation / Theory

The vector product (or cross product) of two vectors 𝐀 and 𝐁 is defined as

\[\mathbf{A} \times \mathbf{B} = AB \sin\theta \, \hat{n}\]

where θ is the smaller angle between them, and \(\hat{n}\) is a unit vector perpendicular to the plane containing 𝐀 and 𝐁, determined by the right-hand rule.
The cross product gives a vector perpendicular to both 𝐀 and 𝐁.

Important Formulas

  • Determinant Form:

    \[\mathbf{A} \times \mathbf{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} = (A_y B_z - A_z B_y)\hat{i} + (A_z B_x - A_x B_z)\hat{j} + (A_x B_y - A_y B_x)\hat{k}\]

  • Magnitude of Cross Product:

    \[|\mathbf{A} \times \mathbf{B}| = AB \sin\theta\]
  • Area Interpretation:
    • The magnitude \(\|\mathbf{A} \times \mathbf{B}\|\) gives the area of the parallelogram formed by 𝐀 and 𝐁.

  • Physical Quantities:

    \(\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F} \quad \text{(Torque)}\) \(\mathbf{L} = \mathbf{r} \times \mathbf{p} \quad \text{(Angular Momentum)}\) \(\mathbf{F} = q(\mathbf{v} \times \mathbf{B}) \quad \text{(Magnetic Force)}\)

Important Points

  • Result is a vector, perpendicular to both 𝐀 and 𝐁.
  • Direction determined by the right-hand rule.
  • Not commutative: \(\mathbf{A} \times \mathbf{B} = -\mathbf{B} \times \mathbf{A}\)
  • Distributive: \(\mathbf{A} \times (\mathbf{B} + \mathbf{C}) = \mathbf{A} \times \mathbf{B} + \mathbf{A} \times \mathbf{C}\)
  • \[\mathbf{A} \times \mathbf{A} = 0\]
  • If 𝐀 and 𝐁 are parallel or antiparallel, then \(\mathbf{A} \times \mathbf{B} = 0\).
  • Used in defining moment of a force, rotational motion, and magnetic interactions.
  • Right-handed triad: The set \((\mathbf{A}, \mathbf{B}, \mathbf{A} \times \mathbf{B})\) forms a right-handed coordinate system.

Scalar Triple Product

Definition / Explanation / Theory

The scalar triple product of three vectors 𝐀, 𝐁, and 𝐂 is defined as

\[\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})\]

It represents the volume of the parallelepiped formed by the three vectors.
The result is a scalar quantity, which can be positive, negative, or zero depending on the orientation of the vectors.

In determinant form:

\[\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = \begin{vmatrix} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \end{vmatrix}\]

Important Formulas

  • Volume of Parallelepiped:

    \[V = |\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})|\]

  • Condition for Coplanarity:

    \[\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = 0\]

Important Points

  • The scalar triple product gives a scalar value equal to the signed volume of the parallelepiped formed by 𝐀, 𝐁, and 𝐂.
  • If the result is zero, the three vectors are coplanar.
  • Cyclic Property:
    \(\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = \mathbf{B} \cdot (\mathbf{C} \times \mathbf{A}) = \mathbf{C} \cdot (\mathbf{A} \times \mathbf{B})\)
  • Anticommutative Property:
    \(\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = -\mathbf{A} \cdot (\mathbf{C} \times \mathbf{B})\)
  • Geometrically, the magnitude represents volume and the sign indicates orientation (right-handed or left-handed system).
  • Physically, if 𝐀, 𝐁, and 𝐂 represent edges of a crystal cell, then
    \(|\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})|\) gives the volume of that cell.

Vector Triple Product

Definition / Explanation / Theory

The vector triple product involves the cross product of one vector with the cross product of two other vectors.
It is defined as

\[\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B}(\mathbf{A} \cdot \mathbf{C}) - \mathbf{C}(\mathbf{A} \cdot \mathbf{B})\]

This identity is known as the vector triple product expansion or BAC–CAB rule.

The result is a vector that lies in the plane of 𝐁 and 𝐂, and is perpendicular to 𝐀 × (𝐁 × 𝐂).

Important Formulas

  • Vector Triple Product Identity: \(\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B}(\mathbf{A} \cdot \mathbf{C}) - \mathbf{C}(\mathbf{A} \cdot \mathbf{B})\)
  • Magnitude Relation:
    Depends on the scalar products \((\mathbf{A} \cdot \mathbf{B})\) and \((\mathbf{A} \cdot \mathbf{C})\).

Important Points

  • The result is not perpendicular to both 𝐁 and 𝐂; it lies within their plane.
  • The operation is not associative, i.e.
    \(\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) \ne (\mathbf{A} \times \mathbf{B}) \times \mathbf{C}\)
  • Commonly used to simplify expressions in electromagnetism, mechanics, and vector field analysis.
  • Example:
    If \(\mathbf{A} = \mathbf{v}, \mathbf{B} = \mathbf{B}, \mathbf{C} = \mathbf{v}\), then
    \(\mathbf{v} \times (\mathbf{B} \times \mathbf{v}) = \mathbf{B}v^2 - \mathbf{v}(\mathbf{v} \cdot \mathbf{B})\)
  • This form frequently appears in the Lorentz force and magnetic motion analysis.

Practice Questions (Easy Level)

1. If

\[\mathbf{A} = 2\hat{i} + 3\hat{j}, \quad \mathbf{B} = \hat{i} + 4\hat{j}\]

then \(\mathbf{A} + \mathbf{B}\) equals
a) \(3\hat{i} + 7\hat{j}\)
b) \(\hat{i} + 7\hat{j}\)
c) \(2\hat{i} + \hat{j}\)
d) \(7\hat{i} + 3\hat{j}\)

2. The magnitude of the resultant of two vectors \(A\) and \(B\) making an angle \(\theta\) is
a) \(\sqrt{A^2 + B^2 + 2AB\sin\theta}\)
b) \(\sqrt{A^2 + B^2 + 2AB\cos\theta}\)
c) \(A + B\cos\theta\)
d) \(A^2 + B^2\)

3. The vector addition follows which law geometrically?
a) Right-hand rule
b) Triangle law or Parallelogram law
c) Law of sines
d) Scalar projection rule

4. The commutative property of vector addition states that
a) \(\mathbf{A} + \mathbf{B} = \mathbf{0}\)
b) \(\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}\)
c) \(\mathbf{A} + \mathbf{B} = -(\mathbf{B} + \mathbf{A})\)
d) \(\mathbf{A} + \mathbf{B} = \mathbf{A}\mathbf{B}\)

5. The scalar (dot) product of two vectors is given by
a) \(AB \tan\theta\)
b) \(AB \cos\theta\)
c) \(AB \sin\theta\)
d) \(A + B\cos\theta\)

6. If \(\mathbf{A} \cdot \mathbf{B} = 0\), the two vectors are
a) Parallel
b) Anti-parallel
c) Perpendicular
d) Collinear

7. The result of a scalar product is
a) Always a vector
b) Always a scalar
c) A matrix
d) Depends on \(\theta\)

8. The work done by a force \(\mathbf{F}\) over a displacement \(\mathbf{d}\) is
a) \(\mathbf{F} \times \mathbf{d}\)
b) \(Fd\sin\theta\)
c) \(\mathbf{F} \cdot \mathbf{d} = Fd\cos\theta\)
d) \(Fd\)

9. The projection of \(\mathbf{A}\) on \(\mathbf{B}\) is
a) \(\mathbf{A} \cdot \mathbf{B}\)
b) \(|\mathbf{A} \times \mathbf{B}|\)
c) \(\dfrac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|}\)
d) \(A + B\)

10. The vector (cross) product of \(\mathbf{A}\) and \(\mathbf{B}\) is
a) \(AB \cos\theta\)
b) \(AB \sin\theta \, \hat{n}\)
c) \(AB \tan\theta\)
d) \(A + B\)

11. The direction of \(\mathbf{A} \times \mathbf{B}\) is determined by
a) Left-hand rule
b) Right-hand rule
c) Fleming’s rule
d) Vector addition rule

12. If two vectors are parallel, then their cross product is
a) Maximum
b) Minimum but nonzero
c) Zero
d) Undefined

13. The magnitude of \(\mathbf{A} \times \mathbf{B}\) represents
a) Area of a triangle formed by \(\mathbf{A}\) and \(\mathbf{B}\)
b) Area of the parallelogram formed by \(\mathbf{A}\) and \(\mathbf{B}\)
c) Volume of the parallelepiped
d) Projection of \(\mathbf{A}\) on \(\mathbf{B}\)

14. The scalar triple product \(\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})\) gives
a) Area of parallelogram
b) Volume of parallelepiped
c) Angle between vectors
d) Torque

15. If \(\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = 0\), then vectors \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{C}\) are
a) Coplanar
b) Collinear
c) Perpendicular
d) Orthogonal

16. The scalar triple product can be written as a determinant of
a) \(2 \times 2\) matrix
b) \(3 \times 3\) matrix
c) \(4 \times 4\) matrix
d) \(1 \times 3\) matrix

17. The cyclic property of scalar triple product is
a) \(\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = \mathbf{B} \cdot (\mathbf{C} \times \mathbf{A})\)
b) \(\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = -\mathbf{A} \cdot (\mathbf{C} \times \mathbf{B})\)
c) Both a and b
d) None of these

18. The vector triple product \(\mathbf{A} \times (\mathbf{B} \times \mathbf{C})\) equals
a) \(\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})\)
b) \(\mathbf{B}(\mathbf{A} \cdot \mathbf{C}) - \mathbf{C}(\mathbf{A} \cdot \mathbf{B})\)
c) \((\mathbf{A} \times \mathbf{B}) \times \mathbf{C}\)
d) None of these

19. The vector triple product \(\mathbf{A} \times (\mathbf{B} \times \mathbf{C})\) lies
a) Along \(\mathbf{A}\)
b) Perpendicular to plane of \(\mathbf{B}\) and \(\mathbf{C}\)
c) In the plane of \(\mathbf{B}\) and \(\mathbf{C}\)
d) Perpendicular to \(\mathbf{A}\) and \(\mathbf{C}\)

20. The operation \(\mathbf{A} \times (\mathbf{B} \times \mathbf{C})\) is
a) Associative
b) Not associative
c) Commutative
d) None of these

Advanced Level Questions for NET/GATE/TIFR

1. If \(\vec{A} = (2\hat{i} + 3\hat{j} - \hat{k})\) and \(\vec{B} = (4\hat{i} - \hat{j} + 2\hat{k})\), then the projection of \(\vec{A}\) on \(\vec{B}\) is:
(a) \(\dfrac{\vec{A} \cdot \vec{B}}{|\vec{B}|}\)
(b) \(\dfrac{\vec{B} \cdot \vec{A}}{|\vec{A}|}\)
(c) \(\dfrac{\vec{A} \times \vec{B}}{|\vec{B}|}\)
(d) \(\dfrac{\vec{A} \cdot \vec{B}}{|\vec{A}|}\)

2. Let \(\vec{F} = (x^2 - y^2)\hat{i} + 2xy\hat{j}\) be a vector field. Which one of the following statements is true?
(a) \(\nabla \cdot \vec{F} = 0\) and \(\nabla \times \vec{F} = 0\)
(b) \(\nabla \cdot \vec{F} = 0\) but \(\nabla \times \vec{F} \neq 0\)
(c) \(\nabla \cdot \vec{F} \neq 0\) but \(\nabla \times \vec{F} = 0\)
(d) Both \(\nabla \cdot \vec{F}\) and \(\nabla \times \vec{F}\) are nonzero

3. Evaluate the line integral \(\oint_C (y^2 dx + x^2 dy)\) where \(C\) is the circle \(x^2 + y^2 = a^2\) taken counterclockwise.
(a) \(0\)
(b) \(\pi a^2\)
(c) \(2\pi a^2\)
(d) \(4\pi a^2\)

4. The vector field \(\vec{A} = (y\hat{i} + x\hat{j})\) is:
(a) Solenoidal and irrotational
(b) Solenoidal but not irrotational
(c) Irrotational but not solenoidal
(d) Neither solenoidal nor irrotational

5. If \(\vec{A} = (yz)\hat{i} + (zx)\hat{j} + (xy)\hat{k}\), then the divergence of \(\vec{A}\) is:
(a) \(x + y + z\)
(b) \(0\)
(c) \(2(x + y + z)\)
(d) \(3xyz\)