1.A quantity that has magnitude as well as direction is called a ________.
2.A quantity that has only magnitude is called a ________.
3.Identify whether 'Temperature' is a scalar or a vector.
4.Identify whether 'Velocity' is a scalar or a vector.
5.Write the formula for the magnitude of vector $\vec{a} = x\hat{i} + y\hat{j} + z\hat{k}$.
6.Find the magnitude of the vector $\vec{a} = 3\hat{i} + 4\hat{j}$.
7.Find the magnitude of the vector $\vec{b} = \hat{i} + \hat{j} + \hat{k}$.
8.What is the position vector of a point $P(x, y, z)$ with respect to the origin $O(0,0,0)$?
9.What is a vector whose magnitude is $1$ called?
10.Write the formula to find the unit vector $\hat{a}$ in the direction of vector $\vec{a}$.
11.Find the unit vector in the direction of $\vec{a} = 3\hat{i} + 4\hat{j}$.
12.Vectors having the same initial point are called ________ vectors.
13.Vectors which are parallel to the same line, irrespective of their magnitudes and directions, are called ________ vectors.
14.If $\vec{a} = x\hat{i} + 2\hat{j} - z\hat{k}$ and $\vec{b} = 3\hat{i} + y\hat{j} + \hat{k}$ are equal vectors, find the values of $x, y, z$.
15.Write the vector $\vec{AB}$ joining the points $A(1, 2, 3)$ and $B(4, 5, 6)$.
16.If a vector makes angles $\alpha, \beta, \gamma$ with the $x, y, z$ axes respectively, what are $\cos\alpha, \cos\beta, \cos\gamma$ called?
17.Write the standard relation between direction cosines $l, m, n$: $l^2 + m^2 + n^2 = $ ________.
18.For any vector $\vec{r} = a\hat{i} + b\hat{j} + c\hat{k}$, what are $a, b, c$ known as?
19.If direction ratios of a vector are proportional to $1, 2, 2$, find its direction cosines.
20.Write the section formula for the position vector of a point $R$ dividing $P(\vec{p})$ and $Q(\vec{q})$ internally in the ratio $m:n$.
21.Write the section formula for the position vector of a point $R$ dividing $P(\vec{p})$ and $Q(\vec{q})$ externally in the ratio $m:n$.
22.Write the midpoint formula for vectors joining $P(\vec{p})$ and $Q(\vec{q})$.
23.Find the midpoint of the vector joining $A(2\hat{i} + 3\hat{j})$ and $B(4\hat{i} + 5\hat{j})$.
24.Write the defining formula for the dot product $\vec{a} \cdot \vec{b}$ in terms of magnitudes and angle $\theta$.
25.Is the result of a dot product ($\vec{a} \cdot \vec{b}$) a scalar or a vector?
26.What is the value of $\hat{i} \cdot \hat{i}$ ?
27.What is the value of $\hat{j} \cdot \hat{j}$ ?
28.What is the value of $\hat{i} \cdot \hat{j}$ ?
29.What is the condition for two non-zero vectors $\vec{a}$ and $\vec{b}$ to be perpendicular (orthogonal)?
30.Write the formula for the angle $\theta$ between two vectors $\vec{a}$ and $\vec{b}$ using the dot product.
31.Write the formula for the scalar projection of vector $\vec{a}$ on vector $\vec{b}$.
32.Find the dot product $\vec{a} \cdot \vec{b}$ if $\vec{a} = 2\hat{i} + \hat{j}$ and $\vec{b} = \hat{i} - 3\hat{j}$.
33.Find the dot product of $(3\hat{i} - 2\hat{j} + \hat{k})$ and $(\hat{i} + \hat{j} - \hat{k})$.
34.True or False: The dot product is commutative, i.e., $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$.
35.What is the scalar projection of $\hat{i}$ on $\hat{j}$?
36.Write the defining formula for the cross product $\vec{a} \times \vec{b}$ involving $\hat{n}$.
37.Is the result of a cross product ($\vec{a} \times \vec{b}$) a scalar or a vector?
38.The direction of $\vec{a} \times \vec{b}$ is determined by which rule?
39.What is the value of $\hat{i} \times \hat{i}$ ?
40.What is the value of $\hat{i} \times \hat{j}$ ?
41.What is the value of $\hat{j} \times \hat{i}$ ?
42.What is the value of $\hat{k} \times \hat{j}$ ?
43.True or False: The cross product is commutative, i.e., $\vec{a} \times \vec{b} = \vec{b} \times \vec{a}$.
44.What is the condition for two non-zero vectors $\vec{a}$ and $\vec{b}$ to be parallel (collinear)?
45.Find the cross product of $2\hat{i}$ and $3\hat{j}$.
46.Write the formula for the Area of a Parallelogram with adjacent sides $\vec{a}$ and $\vec{b}$.
47.Write the formula for the Area of a Triangle with adjacent sides $\vec{a}$ and $\vec{b}$.
48.Write the formula for the Area of a Parallelogram if its diagonals are $\vec{d_1}$ and $\vec{d_2}$.
49.What is the mathematical notation for the Scalar Triple Product (STP) of vectors $\vec{a}, \vec{b}, \vec{c}$?
50.Write the algebraic expansion for the STP: $[\vec{a} \, \vec{b} \, \vec{c}] = \vec{a} \cdot ($ ________ $)$.
51.What geometric figure's volume is represented by the magnitude of $[\vec{a} \, \vec{b} \, \vec{c}]$?
52.What is the necessary condition for three vectors $\vec{a}, \vec{b}, \vec{c}$ to be coplanar?
53.What is the value of the STP if any two vectors inside it are equal, e.g., $[\vec{a} \, \vec{a} \, \vec{b}]$?
54.Find the value of $[\hat{i} \, \hat{j} \, \hat{k}]$.
55.Find the value of $[\hat{i} \, \hat{i} \, \hat{k}]$.
56.Write the BAC-CAB rule for the Vector Triple Product: $\vec{a} \times (\vec{b} \times \vec{c}) = $ ________.
57.Is $(\vec{a} \times \vec{b}) \times \vec{c}$ generally equal to $\vec{a} \times (\vec{b} \times \vec{c})$?
58.Expand $\vec{a} \times (\vec{a} \times \vec{b})$ using the BAC-CAB rule.
59.True or False: The scalar triple product $[\vec{a} \, \vec{b} \, \vec{c}]$ is a scalar quantity.
60.True or False: The vector triple product $\vec{a} \times (\vec{b} \times \vec{c})$ is a vector quantity.
