Assignment Questions

🔷 CHAPTER 1: Vectors and Scalars

Key Definitions

1. Difference Between Scalar and Vector
Definition: unit vector, position vector, scalar field, vector field with example

Q.4 — Find the sum or resultant of the following displacements: A, 10 ft northwest; B, 20 ft 30° north of east; C, 35 ft due south.

Q.21 — Prove that the magnitude A of the vector A = A₁i + A₂j + A₃k is:
$\(A = \sqrt{A_1^2 + A_2^2 + A_3^2}\)$

Q.22 — Given r₁ = 3i − 2j + k, r₂ = 2i − 4j − 3k, r₃ = −i + 2j + 2k, find the magnitudes of:
(a) r₃, (b) r₁ + r₂ + r₃, (c) 2r₁ − 3r₂ − 5r₃

Q.23 — If r₁ = 2i − j + k, r₂ = i + 3j − 2k, r₃ = −2i + j − 3k and r₄ = 3i + 2j + 5k, find scalars a, b, c such that r₄ = ar₁ + br₂ + cr₃.

Q.24 — Find a unit vector parallel to the resultant of vectors r₁ = 2i + 4j − 5k, r₂ = i + 2j + 3k.

Q.25 — Determine the vector having initial point P(x₁, y₁, z₁) and terminal point Q(x₂, y₂, z₂) and find its magnitude.

Q.29 — Given the scalar field φ(x, y, z) = 3x²z − xy³ + 5, find φ at:
(a) (0, 0, 0), (b) (1, −2, 2), (c) (−1, −2, −3)

Q.46 — If a, b, c are non-coplanar vectors, determine whether r₁ = 2a − 3b + c, r₂ = 3a − 5b + 2c, r₃ = 4a − 5b + c are linearly independent or dependent.

Q.58 — The position vectors of points P and Q are r₁ = 2i + 3j − k, r₂ = 4i − 3j + 2k. Determine \(\overrightarrow{PQ}\) in terms of i, j, k and find its magnitude.

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🔷 CHAPTER 2: The Dot and Cross Product

Key Definitions

1. Define The Dot (Scalar) Product and The Cross (Vector) Product

Q.6 — If A = A₁i + A₂j + A₃k and B = B₁i + B₂j + B₃k, prove that:
$\(\mathbf{A} \cdot \mathbf{B} = A_1B_1 + A_2B_2 + A_3B_3\)$

Q.7 — If A = A₁i + A₂j + A₃k, show that:
$\(A = \sqrt{\mathbf{A} \cdot \mathbf{A}} = \sqrt{A_1^2 + A_2^2 + A_3^2}\)$

Q.8 — Find the angle between A = 2i + 2j − k and B = 6i − 3j + 2k.

Q.9 — If A·B = 0 and A, B are not zero vectors, show that A is perpendicular to B.

Q.10 — Determine the value of a so that A = 2i + aj + k and B = 4i − 2j − 2k are perpendicular.

Q.11 — Show that the vectors A = 3i − 2j + k, B = i − 3j + 5k, C = 2i + j − 4k form a right triangle.

Q.12 — Find the angles which the vector A = 3i − 6j + 2k makes with the coordinate axes.

Q.13 — Find the projection of the vector A = i − 2j + k on the vector B = 4i − 4j + 7k.

Q.17 — Find the work done in moving an object along a vector r = 3i + 2j − 5k if the applied force is F = 2i − j − k.

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Solved Problems — Cross Product

Q.27 — If A = A₁i + A₂j + A₃k and B = B₁i + B₂j + B₃k, prove that:
$\(\mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ A_1 & A_2 & A_3 \\ B_1 & B_2 & B_3 \end{vmatrix}\)$

Q.30 — Prove that the area of a parallelogram with sides A and B is |A×B|.

Q.31 — Find the area of the triangle having vertices at P(1, 3, 2), Q(2, −1, 1), R(−1, 2, 3).

Q.32 — Determine a unit vector perpendicular to the plane of A = 2i − 6j − 3k and B = 4i + 3j − k.

Q.37 — Evaluate A·(B×C) (scalar triple product) for given vectors.

Q.38 — Prove that:
$\(\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = \begin{vmatrix} A_1 & A_2 & A_3 \\ B_1 & B_2 & B_3 \\ C_1 & C_2 & C_3 \end{vmatrix}\)$

Q.39 — Find \((2\mathbf{i} - 3\mathbf{j}) \cdot [(\mathbf{i} + \mathbf{j} - \mathbf{k}) \times (3\mathbf{i} - \mathbf{k})]\).

Q.40 — Prove that A·(B×C) = B·(C×A) = C·(A×B).

Q.42 — Prove that A·(A×C) = 0.

Q.43 — Prove that a necessary and sufficient condition for vectors A, B, C to be coplanar is:
$\(\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = 0\)$

Q.47 — Prove:
(a) \(\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B}(\mathbf{A} \cdot \mathbf{C}) - \mathbf{C}(\mathbf{A} \cdot \mathbf{B})\)
(b) \((\mathbf{A} \times \mathbf{B}) \times \mathbf{C} = \mathbf{B}(\mathbf{A} \cdot \mathbf{C}) - \mathbf{A}(\mathbf{B} \cdot \mathbf{C})\)

Q.49 — Prove that:
$\(\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) + \mathbf{B} \times (\mathbf{C} \times \mathbf{A}) + \mathbf{C} \times (\mathbf{A} \times \mathbf{B}) = \mathbf{0}\)$

Q.57 — Find the angle between:
(a) A = 3i + 2j − 6k and B = 4i − 3j + k
(b) C = 4i − 2j + 4k and D = 3i − 6j − 2k

Q.58 — For what values of a are A = ai − 2j + k and B = 2ai + aj − 4k perpendicular?

Q.59 — Find the acute angles which the line joining (1, −3, 2) and (3, −5, 1) makes with the coordinate axes.

Q.60 — Find the direction cosines of the line joining (3, 2, −4) and (1, −1, 2).

Q.61 — Two sides of a triangle are formed by A = 3i + 6j − 2k and B = 4i − j + 3k. Determine the angles of the triangle.

Q.63 — Find the projection of 2i − 3j + 6k on the vector i + 2j + 2k.

Q.64 — Find the projection of 4i − 3j + k on the line passing through (2, 3, −1) and (−2, −4, 3).

Q.65 — If A = 4i − j + 3k and B = −2i + j − 2k, find a unit vector perpendicular to both A and B.

Q.69 — Find the work done in moving an object along a straight line from (3, 2, −1) to (2, −1, 4) in a force field F = 4i − 3j + 2k.

Q.83 — Find the area of a triangle with vertices at (3, −1, 2), (1, −1, −3), (4, −3, 1).

Q.86 — A force F = 3i + 2j − 4k is applied at (1, −1, 2). Find the moment of F about the point (2, −1, 3).

Q.87 — The angular velocity of a rotating rigid body is ω = 4i + j − 2k. Find the linear velocity of point P with position vector 2i − 3j + k relative to a point on the axis of rotation.

Q.90 — Find the volume of the parallelepiped with edges A = 2i − 3j + 4k, B = i + 2j − k, C = 3i − j + 2k.

Q.92 — Find the constant a such that 2i − j + k, i + 2j − 3k, and 3i + aj + 5k are coplanar.

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🔷 CHAPTER 3: Vector Differentiation

Q.1 — If R(u) = x(u)i + y(u)j + z(u)k, where x, y and z are differentiable functions of a scalar u, prove that:
$\(\frac{d\mathbf{R}}{du} = \frac{dx}{du}\mathbf{i} + \frac{dy}{du}\mathbf{j} + \frac{dz}{du}\mathbf{k}\)$

Q.2 — Given R = sin t i + cos t j + tk, find:
$\((a)\ \frac{d\mathbf{R}}{dt}, \quad (b)\ \frac{d^2\mathbf{R}}{dt^2}, \quad (c)\ \left|\frac{d\mathbf{R}}{dt}\right|, \quad (d)\ \left|\frac{d^2\mathbf{R}}{dt^2}\right|\)$

Q.3 — A particle moves along a curve with parametric equations \(x = e^{-t}\), \(y = 2\cos 3t\), \(z = 2\sin 3t\), where t is time.
(a) Determine its velocity and acceleration at any time.
(b) Find the magnitudes of the velocity and acceleration at t = 0.

Q.4 — A particle moves along the curve \(x = 2t^2\), \(y = t^2 - 4t\), \(z = 3t - 5\), where t is time. Find the components of its velocity and acceleration at time t = 1 in the direction i − 3j + 2k.

Q.8 — If A = 5t²i + tj − t³k and B = sin t i − cos t j, find:
$\((a)\ \frac{d}{dt}(\mathbf{A} \cdot \mathbf{B}), \quad (b)\ \frac{d}{dt}(\mathbf{A} \times \mathbf{B}), \quad (c)\ \frac{d}{dt}(\mathbf{A} \cdot \mathbf{A})\)$

Q.12 — A particle moves so that its position vector is r = cosωt i + sinωt j, where ω is a constant. Show that:
(a) The velocity v is perpendicular to r
(b) The acceleration a is directed toward the origin and has magnitude proportional to the distance from the origin
(c) r×v = a constant vector

Q.15 — If \(\mathbf{A} = (2x^2y - x^4)\mathbf{i} + (e^{xy} - y\sin x)\mathbf{j} + (x^2\cos y)\mathbf{k}\), find:
$\(\frac{\partial \mathbf{A}}{\partial x},\ \frac{\partial \mathbf{A}}{\partial y},\ \frac{\partial^2 \mathbf{A}}{\partial x^2},\ \frac{\partial^2 \mathbf{A}}{\partial y^2},\ \frac{\partial^2 \mathbf{A}}{\partial x \partial y},\ \frac{\partial^2 \mathbf{A}}{\partial y \partial x}\)$

Q.16 — If \(\phi(x,y,z) = xy^2z\) and \(\mathbf{A} = xz\mathbf{i} - xy^2\mathbf{j} + yz^2\mathbf{k}\), find \(\dfrac{\partial^2}{\partial x^2 \partial z}(\phi\mathbf{A})\) at the point (2, −1, 1).

Q.32 — Find the velocity and acceleration of a particle moving along \(x = 2\sin 3t\), \(y = 2\cos 3t\), \(z = 8t\) at any time t > 0. Find the magnitudes of velocity and acceleration.

Q.35 — If \(\mathbf{A} = \sin u\,\mathbf{i} + \cos u\,\mathbf{j} + u\mathbf{k}\), \(\mathbf{B} = \cos u\,\mathbf{i} - \sin u\,\mathbf{j} - 3\mathbf{k}\), and \(\mathbf{C} = 2\mathbf{i} + 3\mathbf{j} - \mathbf{k}\), find \(\dfrac{d}{du}(\mathbf{A} \times (\mathbf{B} \times \mathbf{C}))\) at u = 0.

Q.37 — If \(\mathbf{A}(t) = 3t^2\mathbf{i} - (t+4)\mathbf{j} + (t^2 - 2t)\mathbf{k}\) and \(\mathbf{B}(t) = \sin t\,\mathbf{i} + 3e^{-t}\mathbf{j} - 3\cos t\,\mathbf{k}\), find \(\dfrac{d^2}{dt^2}(\mathbf{A} \times \mathbf{B})\) at t = 0.

Q.38 — If \(\dfrac{d^2\mathbf{A}}{dt^2} = 6t\mathbf{i} - 24t^2\mathbf{j} + 4\sin t\,\mathbf{k}\), find A given that A = 2i + j and \(\dfrac{d\mathbf{A}}{dt} = -\mathbf{i} - 3\mathbf{k}\) at t = 0.

Q.39 — Show that \(\mathbf{r} = e^{-t}(\mathbf{C}_1\cos 2t + \mathbf{C}_2\sin 2t)\), where C₁ and C₂ are constant vectors, is a solution of:
$\(\frac{d^2\mathbf{r}}{dt^2} + 2\frac{d\mathbf{r}}{dt} + 5\mathbf{r} = \mathbf{0}\)$

Q.41 — Solve:
$\((a)\ \frac{d^2\mathbf{r}}{dt^2} - 4\frac{d\mathbf{r}}{dt} - 5\mathbf{r} = \mathbf{0}, \quad (b)\ \frac{d^2\mathbf{r}}{dt^2} + 2\frac{d\mathbf{r}}{dt} + \mathbf{r} = \mathbf{0}, \quad (c)\ \frac{d^2\mathbf{r}}{dt^2} + 4\mathbf{r} = \mathbf{0}\)$

Q.44 — If \(\mathbf{A} = x^2yz\,\mathbf{i} - 2xz^3\mathbf{j} + xz^2\mathbf{k}\) and \(\mathbf{B} = 2z\mathbf{i} + y\mathbf{j} - x^2\mathbf{k}\), find \(\dfrac{\partial^2}{\partial x\,\partial y}(\mathbf{A} \times \mathbf{B})\) at (1, 0, −2).