Questions

1. Question 1

   Find the cosines of the angle $\theta$ between $\mathbf{u}$ and $\mathbf{v}$:

   (i)

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

   (ii)

   $\mathbf{u} = [-3, 2, 5]$, $\mathbf{v} = [1, 6, -2]$

   SolutionTheory
2. Question 2

   If $\mathbf{a} + \mathbf{b} + \mathbf{c} = 0$ and $|\mathbf{a}| = 3$, $|\mathbf{b}| = 5$ and $|\mathbf{c}| = 7$. Find the angle between $\mathbf{a}$ and $\mathbf{b}$.

   SolutionTheory
3. Question 3

   If $|\mathbf{a}| = 3$, $|\mathbf{b}| = 4$ and $|\mathbf{a} + \mathbf{b}| = 5$. Find the angle between $\mathbf{a}$ and $\mathbf{b}$.

   SolutionTheory
4. Question 4

   Calculate the projection of $\mathbf{a}$ along $\mathbf{b}$ and projection of $\mathbf{b}$ along $\mathbf{a}$ when:

   (i)

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

   (ii)

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

   SolutionTheory
5. Question 5

   Find a real number $\alpha$ so that the vectors $\mathbf{u}$ and $\mathbf{v}$ are perpendicular:

   (i)

   $\mathbf{u} = \alpha\mathbf{i} + 3\mathbf{j} + \mathbf{k}$, $\mathbf{v} = \mathbf{i} - 2\mathbf{j} + \alpha\mathbf{k}$

   (ii)

   $\mathbf{u} = \alpha\mathbf{i} + 2\alpha\mathbf{j} - \mathbf{k}$, $\mathbf{v} = \mathbf{i} + \alpha\mathbf{j} + 3\mathbf{k}$

   SolutionTheory
6. Question 6

   Find the number $z$ so that the triangle with vertices $A(3, 0, -2)$, $B(0, 3, 1)$ and $C(1, 1, z)$ is a right triangle with right angle at $C$.

   SolutionTheory
7. Question 7

   If $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$ are unit vectors and $2\theta$ is the angle between them, show that $\sin\theta = \dfrac{1}{2}|\hat{\mathbf{a}} - \hat{\mathbf{b}}|$.

   SolutionTheory
8. Question 8

   If $|\mathbf{a} + \mathbf{b}| = |\mathbf{a} - \mathbf{b}|$, then show that $\mathbf{a}$ and $\mathbf{b}$ are perpendicular.

   SolutionTheory
9. Question 9

   (i)

   Show that the vectors $3\mathbf{i} - 2\mathbf{j} + \mathbf{k}$, $\mathbf{i} - 3\mathbf{j} + 5\mathbf{k}$ and $2\mathbf{i} + \mathbf{j} - 4\mathbf{k}$ form a right triangle.

   (ii)

   Show that the set of points $P(4, -1, 2)$, $Q(1, 3, -1)$ and $R(-2, 4, 6)$ form a right triangle.

   SolutionTheory
10. Question 10

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

    SolutionTheory
11. Question 11

    Prove that in any triangle $ABC$.

    (i)

    $b = c\cos A + a\cos C$

    (ii)

    $c = a\cos B + b\cos A$

    (iii)

    $b^2 = c^2 + a^2 - 2ca\cos B$

    (iv)

    $c^2 = a^2 + b^2 - 2ab\cos C$

    SolutionTheory
12. Question 12

    Show that for any vectors $\mathbf{a}$ and $\mathbf{b}$, $||\mathbf{a}| - |\mathbf{b}|| \le |\mathbf{a} + \mathbf{b}| \le |\mathbf{a}| + |\mathbf{b}|$.

    SolutionTheory
13. Question 13

    Find the work done, if the point at which the constant force $\mathbf{F} = 2\mathbf{i} + 5\mathbf{j} + 3\mathbf{k}$ is applied to an object, moves it from $P_1(2, -3, 1)$ to $P_2(7, 5, 3)$.

    SolutionTheory
14. Question 14

    A particle, acted by constant forces $\mathbf{F}_1 = 3\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}$ and $\mathbf{F}_2 = \mathbf{i} + 4\mathbf{j} - \mathbf{k}$, is displaced from $A(2, 1, 3)$ to $B(3, 4, 4)$. Find the work done.

    SolutionTheory
15. Question 15

    A particle is displaced from the point $A(3, -5, -7)$ to the point $B(6, 2, -2)$ under the action of constant forces defined by $10\mathbf{i} - \mathbf{j} + 11\mathbf{k}$, $4\mathbf{i} + 5\mathbf{j} + 9\mathbf{k}$ and $-2\mathbf{i} + \mathbf{j} - 9\mathbf{k}$. Show that the total work done by the force is 102 units.

    SolutionTheory
16. Question 16

    A force of magnitude 6 units acting parallel to $4\mathbf{i} + 3\mathbf{j} - \mathbf{k}$ displace the point of application from $A(2, -1, 3)$ to $B(7, 3, 2)$. Find the work done.

    SolutionTheory