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Assignment Questions

CHAPTER 1: Vectors and Scalars

Key Definitions

  1. Difference Between Scalar and Vector
  2. Definitions with Examples: * Unit vector
    • Position vector
    • Scalar field
    • Vector field

Problems

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 \(A = A_1 i + A_2 j + A_3 k\) is:
$\(A = \sqrt{A_1^2 + A_2^2 + A_3^2}\)$

Q.22 Given \(r_1 = 3i - 2j + k\), \(r_2 = 2i - 4j - 3k\), and \(r_3 = -i + 2j + 2k\), find the magnitudes of:
(a) \(r_3\)
(b) \(r_1 + r_2 + r_3\)
(c) \(2r_1 - 3r_2 - 5r_3\)

Q.23 If \(r_1 = 2i - j + k\), \(r_2 = i + 3j - 2k\), \(r_3 = -2i + j - 3k\), and \(r_4 = 3i + 2j + 5k\), find scalars \(a, b, c\) such that \(r_4 = ar_1 + br_2 + cr_3\).

Q.24 Find a unit vector parallel to the resultant of vectors \(r_1 = 2i + 4j - 5k\) and \(r_2 = i + 2j + 3k\).

Q.25 Determine the vector having initial point \(P(x_1, y_1, z_1)\) and terminal point \(Q(x_2, y_2, z_2)\) and find its magnitude.

Q.29 Given the scalar field \(\phi(x, y, z) = 3x^2z - xy^3 + 5\), find \(\phi\) 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 \(r_1 = 2a - 3b + c\), \(r_2 = 3a - 5b + 2c\), and \(r_3 = 4a - 5b + c\) are linearly independent or dependent.

Q.58 The position vectors of points \(P\) and \(Q\) are \(r_1 = 2i + 3j - k\) and \(r_2 = 4i - 3j + 2k\). Determine \(\vec{PQ}\) in terms of \(i, j, k\) and find its magnitude.

CHAPTER 2: The Dot and Cross Product

Key Definitions

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

Dot Product Problems

Q.6 If \(A = A_1 i + A_2 j + A_3 k\) and \(B = B_1 i + B_2 j + B_3 k\), prove that:
$\(A \cdot B = A_1 B_1 + A_2 B_2 + A_3 B_3\)$

Q.7 If \(A = A_1 i + A_2 j + A_3 k\), show that:
$\(A = \sqrt{A \cdot A} = \sqrt{A_1^2 + A_2^2 + A_3^2}\)$

Q.8 Find the angle between \(A = 2i + 2j - k\) and \(B = 6i - 3j + 2k\).

Q.9 If \(A \cdot B = 0\) 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 \(A = 3i - 2j + k\), \(B = i - 3j + 5k\), and \(C = 2i + j - 4k\) 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 \(F = 2i - j - k\).

Cross Product Problems

Q.27 Prove the determinant form of the cross product:
$\(A \times B = \begin{vmatrix} i & j & k \\ A_1 & A_2 & A_3 \\ B_1 & B_2 & B_3 \end{vmatrix}\)$

Q.30 Prove that the area of a parallelogram with sides \(A\) and \(B\) is \(|A \times B|\).

Q.31 Find the area of the triangle having vertices at \(P(1, 3, 2)\), \(Q(2, -1, 1)\), and \(R(-1, 2, 3)\).

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

Q.37 Evaluate \(A \cdot (B \times C)\) (scalar triple product) for given vectors.

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

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

Q.40 Prove that \(A \cdot (B \times C) = B \cdot (C \times A) = C \cdot (A \times B)\).

Q.42 Prove that \(A \cdot (A \times C) = 0\).

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

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

Q.49 Prove that: \(A \times (B \times C) + B \times (C \times A) + C \times (A \times B) = 0\).

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 \((3, -1, 2)\), \((1, -1, -3)\), and \((4, -3, 1)\).

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 \(\omega = 4i + j - 2k\). 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 \(A = 2i - 3j + 4k\), \(B = i + 2j - k\), and \(C = 3i - j + 2k\).

Q.92 Find the constant \(a\) such that \(2i - j + k\), \(i + 2j - 3k\), and \(3i + aj + 5k\) are coplanar.

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:
$\(\frac{dR}{du} = \frac{dx}{du}i + \frac{dy}{du}j + \frac{dz}{du}k\)$

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

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 \(A = 5t^2i + tj - t^3k\) and \(B = \sin t \, i - \cos t \, j\), find:
(a) \(\frac{d}{dt}(A \cdot B)\)
(b) \(\frac{d}{dt}(A \times B)\)
(c) \(\frac{d}{dt}(A \cdot A)\)

Q.12 A particle moves so that its position vector is \(r = \cos \omega t \, i + \sin \omega t \, j\), where \(\omega\) 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) \(r \times v = \text{a constant vector}\).

Q.15/16 If \(A = (2x^2y - x^4)i + (e^{xy} - y \sin x)j + (x^2 \cos y)k\), find:
$\(\frac{\partial A}{\partial x}, \frac{\partial A}{\partial y}, \frac{\partial^2 A}{\partial x^2}, \frac{\partial^2 A}{\partial y^2}, \frac{\partial^2 A}{\partial x \partial y}, \frac{\partial^2 A}{\partial y \partial x}\)$

Q.16 (cont) If \(\phi(x, y, z) = xy^2z\) and \(A = xzi - xy^2j + yz^2k\), find \(\frac{\partial^3}{\partial x^2 \partial z}(\phi A)\) 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 \(A = \sin u \, i + \cos u \, j + uk\), \(B = \cos u \, i - \sin u \, j - 3k\), and \(C = 2i + 3j - k\), find \(\frac{d}{du}(A \times (B \times C))\) at \(u = 0\).

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

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

Q.39 Show that \(r = e^{-t}(C_1 \cos 2t + C_2 \sin 2t)\), where \(C_1\) and \(C_2\) are constant vectors, is a solution of:
$\(\frac{d^2r}{dt^2} + 2\frac{dr}{dt} + 5r = 0\)$

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

Q.44 If \(A = x^2yzi - 2xz^3j + xz^2k\) and \(B = 2zi + yj - x^2k\), find \(\frac{\partial^2}{\partial x \partial y}(A \times B)\) at \((1, 0, -2)\).