1.Find the magnitude of the vector $\vec{a} = 2\hat{i} - 3\hat{j} + 6\hat{k}$.
2.Find the magnitude of the vector $\vec{b} = \frac{1}{\sqrt{3}}\hat{i} + \frac{1}{\sqrt{3}}\hat{j} - \frac{1}{\sqrt{3}}\hat{k}$.
3.Find the unit vector in the direction of the vector $\vec{a} = \hat{i} + 2\hat{j} + 2\hat{k}$.
4.Find a vector of magnitude 5 in the direction of the vector $3\hat{i} - 4\hat{j}$.
5.If $\vec{a} = 2\hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = \hat{i} - \hat{j} + 2\hat{k}$, find the vector $\vec{a} + \vec{b}$.
6.Find the vector $2\vec{a} - \vec{b}$ using the vectors from the previous question.
7.If the position vectors of points $P$ and $Q$ are $\hat{i} + 2\hat{j} + 3\hat{k}$ and $4\hat{i} + 5\hat{j} + 6\hat{k}$ respectively, find the vector $\vec{PQ}$.
8.Find the magnitude of the vector $\vec{PQ}$ calculated in the previous question.
9.Are the vectors $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$ and $\vec{b} = -2\hat{i} + 4\hat{j} - 6\hat{k}$ collinear? Justify.
10.Find the values of $x$ and $y$ so that the vectors $x\hat{i} + 3\hat{j}$ and $4\hat{i} + y\hat{j}$ are equal.
11.Find the position vector of the midpoint of the line segment joining the points $(2,3,4)$ and $(4,1,-2)$.
12.Show that the vectors $\vec{a} = \hat{i} + \hat{j}$ and $\vec{b} = 2\hat{i} + 2\hat{j}$ have the same direction.
13.Find a vector in the direction of vector $\vec{a} = \hat{i} - 2\hat{j}$ that has a magnitude of 7 units.
14.Find the unit vector in the direction of the sum of the vectors $\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k}$ and $\vec{b} = -\hat{i} + \hat{j} + \hat{k}$.
15.If $\vec{a} = \hat{i} + \hat{j}$ and $\vec{b} = \hat{j} + \hat{k}$, what is the value of $|\vec{a} + \vec{b}|$?
16.Find the direction ratios of the vector $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$.
17.Find the direction cosines of the vector $\vec{a} = \hat{i} + 2\hat{j} + 2\hat{k}$.
18.If a line makes equal angles with the $x, y,$ and $z$ axes, find its direction cosines.
19.Write the direction cosines of the x-axis.
20.Write the direction cosines of the y-axis.
21.Find the position vector of a point $R$ which divides the line joining two points $P(\hat{i} + 2\hat{j} - \hat{k})$ and $Q(-\hat{i} + \hat{j} + \hat{k})$ internally in the ratio 2:1.
22.Find the position vector of a point $R$ which divides the line joining the same points externally in the ratio 2:1.
23.Find the direction cosines of the vector joining the points $A(1, 2, -3)$ and $B(-1, -2, 1)$, directed from $A$ to $B$.
24.If $l, m, n$ are the direction cosines of a vector and $l = \frac{1}{2}, m = \frac{1}{2}$, find the possible values of $n$.
25.Can a vector have direction angles $45^\circ, 60^\circ, 120^\circ$?
26.Find the position vector of the midpoint of $P(2\hat{i} - \hat{j} + 3\hat{k})$ and $Q(4\hat{i} + \hat{j} - \hat{k})$.
27.What are the direction ratios of the line passing through the points $(2, -1, 3)$ and $(3, 1, 5)$?
28.A vector $\vec{r}$ has a magnitude of 14 and direction ratios proportional to $2, 3, -6$. Find the direction cosines of $\vec{r}$.
29.Hence, find the components of the vector $\vec{r}$ from the previous question.
30.Is it possible for a non-zero line/vector to have direction ratios $0, 0, 0$?
31.Find $\vec{a} \cdot \vec{b}$ if $\vec{a} = 2\hat{i} + \hat{j} - 3\hat{k}$ and $\vec{b} = \hat{i} - 2\hat{j} + \hat{k}$.
32.Find the angle between the vectors $\vec{a} = \hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = \hat{i} - \hat{j} + \hat{k}$.
33.Find the scalar projection of the vector $\vec{a} = 2\hat{i} + 3\hat{j} + 2\hat{k}$ on the vector $\vec{b} = \hat{i} + 2\hat{j} + \hat{k}$.
34.For what value of $\lambda$ are the vectors $\vec{a} = 2\hat{i} + \lambda\hat{j} + \hat{k}$ and $\vec{b} = \hat{i} - 2\hat{j} + 3\hat{k}$ perpendicular to each other?
35.Evaluate $(\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b})$ given that $|\vec{a}| = 5$ and $|\vec{b}| = 3$.
36.Find $|\vec{a} - \vec{b}|$ if $|\vec{a}| = 2$, $|\vec{b}| = 3$ and $\vec{a} \cdot \vec{b} = 4$.
37.If $\vec{a}$ is a unit vector and $(\vec{x} - \vec{a}) \cdot (\vec{x} + \vec{a}) = 8$, find the magnitude of $\vec{x}$ ($|\vec{x}|$).
38.Find the value of the expression: $\hat{i} \cdot (\hat{j} + \hat{k}) + \hat{j} \cdot (\hat{i} + \hat{k}) + \hat{k} \cdot (\hat{i} + \hat{j})$.
39.Find the scalar projection of $\hat{i} - \hat{j}$ on $\hat{i} + \hat{j}$.
40.Determine if the vectors $\hat{i} - 2\hat{j} + 5\hat{k}$ and $-2\hat{i} + 4\hat{j} + 2\hat{k}$ are orthogonal.
41.If $\vec{a}$ and $\vec{b}$ are unit vectors and $\theta$ is the angle between them, find $\cos\theta$ if $\vec{a} \cdot \vec{b} = 1/2$.
42.Write the vector projection of $\vec{a}$ on $\vec{b}$ in formula form (not scalar projection).
43.If $\vec{a} \cdot \vec{a} = 0$, what can you say about the vector $\vec{a}$?
44.Expand and evaluate $(3\vec{a} - 5\vec{b}) \cdot (2\vec{a} + 7\vec{b})$.
45.Find the angle between the x-axis and the vector $\hat{i} + \hat{j} + \hat{k}$.
46.Find the cross product $\vec{a} \times \vec{b}$ if $\vec{a} = 2\hat{i} + \hat{j} + 3\hat{k}$ and $\vec{b} = 3\hat{i} + 5\hat{j} - 2\hat{k}$.
47.Find the magnitude of $\vec{a} \times \vec{b}$ from the previous question.
48.Find a unit vector perpendicular to both the vectors $\vec{a} = \hat{i} + \hat{j}$ and $\vec{b} = \hat{j} + \hat{k}$.
49.Find the area of a parallelogram whose adjacent sides are determined by the vectors $\vec{a} = \hat{i} - \hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} - 7\hat{j} + \hat{k}$.
50.Find the area of a triangle having vertices at points $A(1, 1, 1), B(1, 2, 3)$, and $C(2, 3, 1)$.
51.For what values of $p$ and $q$ are the vectors $2\hat{i} + 3\hat{j} - \hat{k}$ and $p\hat{i} + q\hat{j} - 3\hat{k}$ parallel?
52.Evaluate the expression: $\hat{i} \times (\hat{j} + \hat{k}) + \hat{j} \times (\hat{k} + \hat{i}) + \hat{k} \times (\hat{i} + \hat{j})$.
53.If $|\vec{a}| = 2$, $|\vec{b}| = 5$ and $|\vec{a} \times \vec{b}| = 8$, find the value of $\vec{a} \cdot \vec{b}$.
54.Find the area of the parallelogram whose diagonals are given by $\vec{d_1} = 3\hat{i} + \hat{j} - 2\hat{k}$ and $\vec{d_2} = \hat{i} - 3\hat{j} + 4\hat{k}$.
55.Compute the cross product: $(\hat{i} - \hat{j}) \times (\hat{i} + \hat{j})$.
56.Let $\vec{a}, \vec{b}$ be two non-zero vectors. If $\vec{a} \times \vec{b} = \vec{0}$, what is the angle between them?
57.Calculate the cross product of $\vec{a} = 3\hat{i} - 4\hat{j}$ and $\vec{b} = -2\hat{i} + 3\hat{k}$.
58.Prove using distributive properties that $(\vec{a} - \vec{b}) \times (\vec{a} + \vec{b}) = 2(\vec{a} \times \vec{b})$.
59.What is the value of the cross product of a vector with itself, i.e., $\vec{a} \times \vec{a}$?
60.If $\vec{a} = 2\hat{i} + 3\hat{j}$, find the unit vector parallel to the z-axis that is perpendicular to $\vec{a}$.
61.Find the scalar triple product $[\vec{a} \, \vec{b} \, \vec{c}]$ if $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$, $\vec{b} = 2\hat{i} + \hat{j} - \hat{k}$, and $\vec{c} = \hat{j} + \hat{k}$.
62.Determine if the vectors $\vec{a} = \hat{i} + 3\hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} - \hat{j} - \hat{k}$, and $\vec{c} = 7\hat{j} + 3\hat{k}$ are coplanar.
63.Find the volume of a parallelepiped whose coterminous edges are $\vec{a} = 2\hat{i} + 3\hat{j} + 4\hat{k}$, $\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$, and $\vec{c} = 3\hat{i} - \hat{j} + 2\hat{k}$.
64.Find the value of $\lambda$ if the vectors $\hat{i} - \hat{j} + \hat{k}$, $3\hat{i} + \hat{j} + 2\hat{k}$, and $\hat{i} + \lambda\hat{j} - 3\hat{k}$ are coplanar.
65.Evaluate the sum: $[\hat{i} \, \hat{j} \, \hat{k}] + [\hat{j} \, \hat{k} \, \hat{i}] + [\hat{k} \, \hat{i} \, \hat{j}]$.
66.Verify conceptually that $\vec{a} \cdot (\vec{a} \times \vec{b}) = 0$.
67.Use the BAC-CAB rule to expand $\hat{i} \times (\hat{j} \times \hat{i})$ and find its value.
68.Find the value of $\hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j})$.
69.If $[\vec{a} \, \vec{b} \, \vec{c}] = 5$, find the value of $[2\vec{a} \, 3\vec{b} \, 4\vec{c}]$.
70.What is the value of $[\vec{a} + \vec{b} \quad \vec{b} + \vec{c} \quad \vec{c} + \vec{a}]$ in terms of $[\vec{a} \, \vec{b} \, \vec{c}]$?
