Questions

1. Question 1

   Let $\mathbf{u} = 3\mathbf{i} + 2\mathbf{j} - 5\mathbf{k}$, $\mathbf{v} = \mathbf{i} - 5\mathbf{j} - \mathbf{k}$ and $\mathbf{w} = -4\mathbf{i} - \mathbf{j} + 7\mathbf{k}$. Find the following:

   (i)

   $\mathbf{u} + 2\mathbf{v} + \mathbf{w}$

   (ii)

   $\mathbf{v} - 3\mathbf{w}$

   (iii)

   $|3\mathbf{v} + \mathbf{w}|$

   SolutionTheory
2. Question 2

   Find the magnitude of the vector $\mathbf{v}$ and write the direction cosines of $\mathbf{v}$.

   (i)

   $\mathbf{v} = 3\mathbf{i} - 2\mathbf{j} + 6\mathbf{k}$

   (ii)

   $\mathbf{v} = -4\mathbf{i} + 4\mathbf{j} + 2\mathbf{k}$

   (iii)

   $\mathbf{v} = -6\mathbf{i} + 8\mathbf{j}$

   SolutionTheory
3. Question 3

   Find $t$, so that $|2\mathbf{i} + (t - 1)\mathbf{j} + t\mathbf{k}| = \sqrt{13}$.

   SolutionTheory
4. Question 4

   Find a unit vector in the direction of $\mathbf{v} = -\mathbf{i} + 4\mathbf{j} - 8\mathbf{k}$.

   SolutionTheory
5. Question 5

   If $\mathbf{u} = 2\mathbf{i} + \mathbf{j} - 3\mathbf{k}$, $\mathbf{v} = -\mathbf{i} + 4\mathbf{j} + 2\mathbf{k}$ and $\mathbf{w} = 3\mathbf{i} - 2\mathbf{j} + \mathbf{k}$, Find a unit vector parallel to $4\mathbf{u} - 3\mathbf{v} + 2\mathbf{w}$.

   SolutionTheory
6. Question 6

   Find a vector whose

   (i)

   magnitude is 5 and is parallel to $3\mathbf{i} + 4\mathbf{j} - \mathbf{k}$

   (ii)

   magnitude is 7 and is parallel to $-\mathbf{i} + \mathbf{j} + \mathbf{k}$

   SolutionTheory
7. Question 7

   If $\mathbf{u} = x\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$, $\mathbf{v} = \mathbf{i} + y\mathbf{j} - 3\mathbf{k}$ and $\mathbf{w} = -2\mathbf{i} - 3\mathbf{j}$ represent the sides of a triangle. Find the values of $x$ and $y$.

   SolutionTheory
8. Question 8

   The position vectors of the points $A, B, C$ and $D$ are $\mathbf{u} = \mathbf{i} + 2\mathbf{j} + \mathbf{k}$, $\mathbf{v} = 7\mathbf{i} + 8\mathbf{j} + 4\mathbf{k}$, $\mathbf{w} = -\mathbf{i} + \mathbf{k}$ and $\mathbf{z} = \mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$ respectively. Show that $\overrightarrow{AB}$ is parallel to $\overrightarrow{CD}$.

   SolutionTheory
9. Question 9

   We say that two vectors $\mathbf{v}$ and $\mathbf{w}$ in space are parallel if there is a scalar $c$ such that $\mathbf{v} = c\mathbf{w}$.

   (a)

   Find two vectors of length 2 parallel to the vector $\mathbf{v} = 2\mathbf{i} - 4\mathbf{j} + 4\mathbf{k}$.

   (b)

   Find the constant $a$ so that the vectors $\mathbf{v} = \mathbf{i} - 3\mathbf{j} + 4\mathbf{k}$ and $\mathbf{w} = a\mathbf{i} + 9\mathbf{j} - 12\mathbf{k}$ are parallel.

   (c)

   Find a vector of length 5 in the direction opposite that of $\mathbf{v} = \mathbf{i} - 2\mathbf{j} + 3\mathbf{k}$.

   (d)

   Find $a$ and $b$ so that the vectors $3\mathbf{i} - \mathbf{j} + 4\mathbf{k}$ and $a\mathbf{i} + b\mathbf{j} - 2\mathbf{k}$ are parallel.

   SolutionTheory
10. Question 10

    A spacecraft moves from point $(120, 240, -50)$ to point $(130, 210, 80)$ in kilometres. What is the magnitude of the displacement vector in kilometres?

    SolutionTheory
11. Question 11

    Find the direction cosines for the given vector:

    (i)

    $\mathbf{u} = -6\mathbf{i} + 3\mathbf{j} + 2\mathbf{k}$

    (ii)

    $\mathbf{v} = 4\mathbf{i} + 2\mathbf{j} - 5\mathbf{k}$

    (iii)

    $\overrightarrow{PQ}$ where $P(9, 3, 13)$ and $Q(11, 6, 19)$

    SolutionTheory
12. Question 12

    Which of the following triple can be the direction angles of a single vector?

    (i)

    $45^\circ, 45^\circ, 60^\circ$

    (ii)

    $30^\circ, 45^\circ, 60^\circ$

    (iii)

    $45^\circ, 60^\circ, 60^\circ$

    SolutionTheory