Questions

1. Question 1

   Find the volume of parallelepiped for which the given vectors are three edges

   (i)

   $\mathbf{u} = 3\mathbf{i} + 2\mathbf{k};\; \mathbf{v} = \mathbf{i} + 2\mathbf{j} + \mathbf{k};\; \mathbf{w} = -\mathbf{j} + 4\mathbf{k}$

   (ii)

   $\mathbf{u} = \mathbf{i} - 4\mathbf{j} - \mathbf{k};\; \mathbf{v} = \mathbf{i} - \mathbf{j} - 2\mathbf{k};\; \mathbf{w} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k}$

   (iii)

   $\mathbf{u} = \mathbf{i} - 2\mathbf{j} + 3\mathbf{k};\; \mathbf{v} = 2\mathbf{i} - \mathbf{j} - \mathbf{k};\; \mathbf{w} = \mathbf{j} + \mathbf{k}$

   SolutionTheory
2. Question 2

   Verify that $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = \mathbf{c} \cdot (\mathbf{a} \times \mathbf{b})$ If $\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 5\mathbf{k};\; \mathbf{b} = 4\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}$ and $\mathbf{c} = 2\mathbf{i} + 5\mathbf{j} + \mathbf{k}$.

   SolutionTheory
3. Question 3

   Prove that the vectors $\mathbf{i} - 2\mathbf{j} + 3\mathbf{k}$, $-2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}$ and $\mathbf{i} - 3\mathbf{j} + 5\mathbf{k}$ are coplanar.

   SolutionTheory
4. Question 4

   Find the constant $\alpha$ such that the vectors are coplanar.

   (i)

   $\mathbf{i} - \mathbf{j} + \mathbf{k}$, $\mathbf{i} - 2\mathbf{j} - 3\mathbf{k}$ and $3\mathbf{i} - \alpha\mathbf{j} + 5\mathbf{k}$

   (ii)

   $\mathbf{i} - 2\alpha\mathbf{j} - \mathbf{k}$, $\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}$ and $\alpha\mathbf{i} - 2\mathbf{j} + \mathbf{k}$

   SolutionTheory
5. Question 5

   Prove that the points whose position vectors are $A(-6\mathbf{i} + 3\mathbf{j} + 2\mathbf{k})$, $B(3\mathbf{i} - 2\mathbf{j} + 4\mathbf{k})$, $C(5\mathbf{i} + 7\mathbf{j} + 3\mathbf{k})$, $D(-13\mathbf{i} + 17\mathbf{j} - \mathbf{k})$ are coplanar.

   SolutionTheory
6. Question 6

   (a)

   Find the value of: (i) $2\mathbf{i} \times 2\mathbf{j} \cdot \mathbf{k}$ (ii) $3\mathbf{j} \cdot (\mathbf{k} \times \mathbf{i})$ (iii) $[\mathbf{k}\,\mathbf{i}\,\mathbf{j}]$ (iv) $[\mathbf{i}\,\mathbf{i}\,\mathbf{k}]$

   (b)

   Prove that $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) + \mathbf{v} \cdot (\mathbf{w} \times \mathbf{u}) + \mathbf{w} \cdot (\mathbf{u} \times \mathbf{v}) = 3\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$

   SolutionTheory
7. Question 7

   Find volume of tetrahedron with the vertices

   (i)

   $(0, 1, 2)$, $(3, 2, 1)$, $(1, 2, 1)$ and $(5, 5, 6)$

   (ii)

   $(2, 1, 8)$, $(3, 2, 9)$, $(2, 1, 4)$ and $(3, 3, 10)$

   SolutionTheory
8. Question 8

   Prove that the points whose position vectors are $A(3\mathbf{i} + 2\mathbf{j} - \mathbf{k})$, $B(\mathbf{i} - 2\mathbf{j} + \mathbf{k})$, $C(6\mathbf{i} + 4\mathbf{j} - 2\mathbf{k})$, $D(9\mathbf{i} + 6\mathbf{j} - 3\mathbf{k})$ are coplanar.

   SolutionTheory
9. Question 9

   Prove that for any three non-zero vectors $\mathbf{u}$, $\mathbf{v}$ and $\mathbf{w}$, $(\mathbf{u} + \mathbf{v}) \cdot [(\mathbf{v} + \mathbf{w}) \times (\mathbf{w} + \mathbf{u})] = 2[\mathbf{u}\,\mathbf{v}\,\mathbf{w}]$.

   SolutionTheory
10. Question 10

    Consider a parallelepiped determined by the vector $\mathbf{u} = 2\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}$, $\mathbf{v} = 5\mathbf{i} - 3\mathbf{j} + 6\mathbf{k}$ and $\mathbf{w} = 4\mathbf{i} - 7\mathbf{j} - 2\mathbf{k}$. If the base of the parallelepiped is define by the vectors $\mathbf{u}$ and $\mathbf{v}$ then find the height of the parallelepiped.

    SolutionTheory
11. Question 11

    A mechanic applies a force of 50 pounds along the positive x-axis on a wrench connected to a bolt. The pivot point of the wrench is at the origin (0, 0, 0), and the force is applied at the point (0 ft, 2 ft, 3 ft). Determine the torque produced by this force about the pivot point.

    SolutionTheory
12. Question 12

    A drone flies from point (1, 2, 5) to point (4, 6, 9), with each unit representing a meter. What is the magnitude of the displacement the drone experienced during this flight?

    SolutionTheory
13. Question 13

    The vector $\mathbf{u} = 50\mathbf{i} + 75\mathbf{j} + 65\mathbf{k}$ shows how many belts, pants, and shirts were sold at a store. The vector $\mathbf{w} = 1500\mathbf{i} + 3500\mathbf{j} + 3000\mathbf{k}$ shows the price (in rupees) of each item. Find $\mathbf{u} \cdot \mathbf{w}$ and explain what the result tells us in real life.

    SolutionTheory
14. Question 14

    A force $\mathbf{F} = (20, -10, 30)\,\mathrm{N}$ is applied at a point $P(2, -1, 4)$ in 3D space. The pivot point is at $M(1, 2, -3)$. Calculate the torque produced by this force about the pivot point $M$.

    SolutionTheory
15. Question 15

    An electric shop sells three types of appliances: Fans, Heaters, and Ovens. The monthly sales quantities are 500 units of Fans, 300 units of Heaters and 200 units of Ovens. The profit per unit for each appliance is Rs 500 for Fans, Rs 400 for Heaters, and Rs 2,000 for Ovens.

    (a)

    Represent the monthly sales quantities and the profit per unit as vectors.

    (b)

    Calculate the total monthly profit using vector operations.

    SolutionTheory