Questions

1. Question 1

   Compute the cross product $\mathbf{a} \times \mathbf{b}$ and $\mathbf{b} \times \mathbf{a}$. Check your answer by showing that each $\mathbf{a}$ and $\mathbf{b}$ are perpendicular to $\mathbf{a} \times \mathbf{b}$ and $\mathbf{b} \times \mathbf{a}$.

   (i)

   $\mathbf{a} = 2\mathbf{i} + \mathbf{j} - \mathbf{k}$, $\mathbf{b} = \mathbf{i} - \mathbf{j} + \mathbf{k}$

   (ii)

   $\mathbf{a} = \mathbf{i} + 3\mathbf{j} + 2\mathbf{k}$, $\mathbf{b} = 2\mathbf{i} - \mathbf{j} + \mathbf{k}$

   (iii)

   $\mathbf{a} = 2\mathbf{i} - 2\mathbf{j} + \mathbf{k}$, $\mathbf{b} = -\mathbf{i} + \mathbf{j} + 3\mathbf{k}$

   (iv)

   $\mathbf{a} = -4\mathbf{i} + \mathbf{j} - 2\mathbf{k}$, $\mathbf{b} = 2\mathbf{i} + \mathbf{j} + \mathbf{k}$

   SolutionTheory
2. Question 2

   Find a unit vector perpendicular to the plane containing $\mathbf{a}$ and $\mathbf{b}$. Also find sine of the angle between them:

   (i)

   $\mathbf{a} = \mathbf{i} + 6\mathbf{j} - 3\mathbf{k}$, $\mathbf{b} = 2\mathbf{i} + \mathbf{j} + 3\mathbf{k}$

   (ii)

   $\mathbf{a} = -\mathbf{i} - \mathbf{j} - \mathbf{k}$, $\mathbf{b} = 2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k}$

   (iii)

   $\mathbf{a} = \mathbf{i} + \mathbf{j} + \mathbf{k}$, $\mathbf{b} = \mathbf{i} - \mathbf{j} - \mathbf{k}$

   (iv)

   $\mathbf{a} = 5\mathbf{i} + \mathbf{j} - 3\mathbf{k}$, $\mathbf{b} = -2\mathbf{i} + 4\mathbf{j} + \mathbf{k}$

   SolutionTheory
3. Question 3

   Find the area of the triangle, formed by the points $P, Q$ and $R$.

   (i)

   $P(2, 3, 5); Q(1, 2, 0); R(4, 1, 2)$

   (ii)

   $P(0, 0, 1); Q(2, -1, 2); R(-1, 3, 2)$

   SolutionTheory
4. Question 4

   Find the area of a parallelogram, whose vertices are:

   (i)

   $A(1, 1, 1); B(4, 2, 3); C(5, 6, 7); D(2, 5, 5)$

   (ii)

   $A(4, 5, 6); B(1, 3, 2); C(-2, 0, 1); D(1, 2, 5)$

   SolutionTheory
5. Question 5

   If the cross product of the vectors $\mathbf{u} = 7\mathbf{i} - 4\mathbf{j} + 5\mathbf{k}$ and $\mathbf{v} = a\mathbf{i} - b\mathbf{j} + 3\mathbf{k}$ is zero, then find the values of $a$ and $b$.

   SolutionTheory
6. Question 6

   Which vectors, if any, are perpendicular or parallel

   (i)

   $\mathbf{u} = 5\mathbf{i} - \mathbf{j} + \mathbf{k}; \mathbf{v} = \mathbf{j} - 5\mathbf{k}; \mathbf{w} = -15\mathbf{i} + 3\mathbf{j} - 3\mathbf{k}$

   (ii)

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

   SolutionTheory
7. Question 7

   Use the definition of cross product, for any vectors $\mathbf{u}, \mathbf{v}, \mathbf{w}$ and scalar $k$, prove that

   (i)

   $\mathbf{u} \times (-\mathbf{u}) = 0$

   (ii)

   $\mathbf{u} \times \mathbf{v} = -\mathbf{v} \times \mathbf{u}$

   (iii)

   $\mathbf{u} \times (k\mathbf{v}) = (k\mathbf{u}) \times \mathbf{v} = k(\mathbf{u} \times \mathbf{v})$

   (iv)

   $\mathbf{u} \times (\mathbf{v} + \mathbf{w}) = (\mathbf{u} \times \mathbf{v}) + (\mathbf{u} \times \mathbf{w})$

   SolutionTheory
8. Question 8

   Prove that: $\mathbf{a} \times (\mathbf{b} + \mathbf{c}) + \mathbf{b} \times (\mathbf{c} + \mathbf{a}) + \mathbf{c} \times (\mathbf{a} + \mathbf{b}) = 0$.

   SolutionTheory
9. Question 9

   If $\mathbf{a} + \mathbf{b} + \mathbf{c} = 0$, then prove that $\mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{c} = \mathbf{c} \times \mathbf{a}$.

   SolutionTheory
10. Question 10

    Prove that: $\sin(\alpha - \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$.

    SolutionTheory
11. Question 11

    Show that $|\mathbf{a} \times \mathbf{b}|^2 = |\mathbf{a}|^2 |\mathbf{b}|^2 - (\mathbf{a} \cdot \mathbf{b})^2$.

    SolutionTheory
12. Question 12

    Use the definition of cross product, prove that for any vectors $\mathbf{u}$ and $\mathbf{v}$ $(\mathbf{u} + \mathbf{v}) \times (\mathbf{u} - \mathbf{v}) = -2(\mathbf{u} \times \mathbf{v})$.

    SolutionTheory
13. Question 13

    Find the moment about the point $M(1, -3, 3)$ of the force represented by $\overrightarrow{AB}$, where the coordinates of points $A(4, 3, -1)$ and $B(-1, 3, 7)$ are given.

    SolutionTheory
14. Question 14

    A force $\mathbf{F} = 6\mathbf{i} + 4\mathbf{j} - 4\mathbf{k}$ is applied at the point $A(1, -1, 2)$. Find the moment of the force about the point $B(3, -2, 3)$.

    SolutionTheory
15. Question 15

    Give a force $\mathbf{F} = 2\mathbf{i} + \mathbf{j} - 3\mathbf{k}$ acting at a point $A(1, -2, 1)$. Find the moment of $\mathbf{F}$ about the point $B(2, 0, -2)$.

    SolutionTheory
16. Question 16

    A force $\mathbf{F} = -2\mathbf{i} + \mathbf{j} - 3\mathbf{k}$ is applied at $P(-1, -3, 2)$. Find its moment about the point $Q(4, 2, 2)$.

    SolutionTheory