Any physical quantity is represented by a "function" in mathematics. Take the example of temperature T. We know that the temperature of the surroundings is changing throughout the day. It increases till noon and decreases in the evening. At any time "t" the temperature T has a unique value. Mathematically this variation can be represented by the notation 'T (t)' and it should be called "temperature as a function of time". It implies that if the value of 't' is given, then the function "T (t)" will give the value of the temperature at that time 't'.
Eg: Consider a function f(x) = x² Sometimes it is also represented as y = x². Here y is called the dependent .variable and x is called independent variable. It means as x changes, y also changes. Once a physical quantity is represented by a function, one can study the variation of the function over time or over the independent variable on which the quantity depends.

Two vectors have same direction.

No.

Time of flight (T_f) The total time taken by the projectile from the point of projection till it hits the horizontal plane is called time of flight.

(i) When the projectile hits the ground after initially thrown horizontally from the top of tower of height h, the time of flight is
\(t=\sqrt { \frac { 2h }{ g } } \)
(ii) The horizontal component velocity of the projectile remains the same i.e V_x = u.
(iii) The vertical component velocity of the projectile at time T is
\({ v }_{ y }=gt=g\sqrt { \frac { 2h }{ g } } =\sqrt { 2gh } \quad \)
(iv) The speed of the particle when it reaches the ground is
\(v=\sqrt { { u }^{ 2 }+2gh } \) 

(i) Air resistance is neglected.
(ii) The effect due to rotation of Earth and curvature of Earth is negligible.
(iii) The acceleration due to gravity is constant in magnitude and direction at all points of the motion of the projectile.

(i) Torque \(\overrightarrow { t } =\overrightarrow { r } \times \overrightarrow { F } \). where F is Force and \(\overrightarrow { r } \) is position vector of a particle
(ii) Angular momentum \(\overrightarrow { L } =\overrightarrow { r } \times \overrightarrow { p }\ where\ \overrightarrow { p } \)  is the linear momentum 
(iii) Linear Velocity \(\overrightarrow { v } =\overrightarrow { \omega } \times \overrightarrow { r } \)  where \(\overrightarrow { \omega } \) is angular velocity.

It is the process of splitting a vector into two or more vectors in such a way that their combined effect is same as that of the given vector.

i) The vector product of two vectors is not commutative, i.e., \(\overrightarrow { A } \times \overrightarrow { B } \neq \overrightarrow { B } \times \overrightarrow { A } \) But., \(\overrightarrow { A } \times \overrightarrow { B } =-[\overrightarrow { B } \times \overrightarrow { A } ]\)
(ii) Here it is worthwhile to note that \(|\overrightarrow { A } \times \overrightarrow { B } |=[\overrightarrow { B } \times \overrightarrow { A } ]\)= AB sin \(\theta\)  i.e., in the case of the product vectors \(\overrightarrow { A } \) \(\times\)\(\overrightarrow { B } \) and \(\overrightarrow { B } \) \(\times\) \(\overrightarrow { A } \) , the magnitudes are equal but directions are opposite to each other.

Max.height = \(\frac { { u }^{ 2 }{ sin }^{ 2 }\theta }{ 2g } \)
\(Range \ R=\frac { { u }^{ 2 }{ sin 2 }\theta }{ 2g } \)
Here, R = 4 x Max.height
\(\therefore \frac { { u }^{ 2 }{ sin 2 }\theta }{ g } =\frac { {4\times u }^{ 2 }{ sin }^{ 2 }\theta }{ 2g } \)
\(sin2\theta =2{ sin }^{ 2 }\theta \)
\(2sin\theta cos\theta =2{ sin }^{ 2 }\theta \)
\(\therefore \ cos\theta = sin \theta\)
\(\therefore \theta =45°\)

(i) A quantity which is described by both its magnitude and direction is called a vector quantity.
(ii) Geometrically, a vector is a directed line segment
Examples: Force, velocity displacement, acceleration, position vector, linear momentum and angular momentum.

The two vectors \(\vec{A}\) and \(\vec{B}\) in a Cartesian coordinate system can be expressed as
\(\overrightarrow { A } ={ A }_{ x }\hat { i } +{ A }_{ y }\hat { j } +{ A }_{ z }\hat { k } \)
\(\overrightarrow { B } ={ B }_{ x }\hat { i } +{ B }_{ y }\hat { j } +{ B }_{ z }\hat { k } \)
Then the addition of two vectors is equivalent to adding their corresponding x, y and z components.
\(\vec{A}+\vec{B}=(A_x+B_x)\hat{i}+(A_y+B_y)\hat{j}+(A_z+B_z)\hat{k}\)

\(\overrightarrow { r } =3\hat { i } +2\hat { j } \)
\(\text { Unit vector } =\frac{\vec{r}}{|\vec{r}|}  \)
\(|\vec{r}| =\sqrt{3^{2}+2^{2}}=\sqrt{13}  \)
\(\therefore \text { Unit vector } =\frac{3 \hat{i}+2 \hat{j}}{\sqrt{13}}\)

It is defined as another vector having a magnitude equal to the product of the magnitudes of two vectors and the sine of the angle between them.
If \(\overrightarrow { A } \) and \(\overrightarrow { B } \) are two vectors, then their vector product \(\vec{A} \times \vec{B}=\vec{C}=(A B \sin \theta) \hat{n}\).
The direction of the product vector \((\hat{n})\) is perpendicular to the plane containing the two vectors, in accordance with the right hand screw rule or right hand thumb rule.
1. It is not commutative, i.e., \(\vec{A} \times \vec{B} \neq \vec{B} \times \vec{A}. \ But \ \vec{A} \times \vec{B}=-[\vec{B} \times \vec{A}].\)
2. \((\vec{A} \times \vec{B})_{\max }=A B \hat{n}\), when \(\theta=90^{\circ}\) i.e., when \(\overrightarrow { A } \) and \(\overrightarrow { B } \) are orthogonal to each other.
3. \((\vec{A} \times \vec{B})_{\text {min }}=0\), when \(\theta=0^{\circ} \ or \ 180^{\circ}\) i.e., when the vectors are either parallel or antiparallel provided \(\overrightarrow { A } \) and \(\overrightarrow { B } \) are non-zero vectors.
4. The self - vector products of unit vectors are \(\hat{i} \times \hat{i}=\hat{j} \times \hat{j}=\hat{k} \times \hat{k}=0\)
5. In the case of orthogonal unit vectors, \(\hat{i} \times \hat{j}=\hat{k}, \hat{j} \times \hat{k}=i, \hat{k} \times \hat{i}=j\)
6. Torque \(\tau=\vec{r} \times \overrightarrow{\mathrm{F}}\), angular momentum \(\vec{L}=\vec{r} \times \vec{p}\ an \ \vec{V}=\vec{w} \times \vec{r}\) are examples of vector product.

It is defined as the product of the magnitudes of both the vectors and the cosine of the angle between them.
If there are two vectors \(\overrightarrow { A } \) and \(\overrightarrow { B } \) having an angle θ between then,  \(\overrightarrow { A } \).\(\overrightarrow { B } \)= ABCDθ.
Here, A and B are magnitude of \(\overrightarrow { A } \) and \(\overrightarrow { B } \).
1. It is commutative, i.e., \(\vec{A} \cdot \vec{B}=\vec{B} \cdot \vec{A}\).
2. It obeys distributive law, i.e, \(\vec{A} \cdot(\vec{B}+\vec{C})=\vec{A} \cdot \vec{B}+\vec{A} \cdot \vec{C}\).
3. \((\vec{A} \cdot \vec{B})_{\max }=\mathrm{AB}\), when \(\theta=0^{\circ}\), i.e., when the vectors are parallel.
4. \((\vec{A} \cdot \vec{B})_{\min }=-A B\), when \(\theta=180^{\circ}\), i.e., when the vectors are antiparallel.
5. \(\vec{A} \cdot \vec{B}=0\) when \(\theta=90^{\circ}\), i.e., when the vectors are mutually orthogonal.
6. The self dot product of unit vectors \(\vec{i}, \vec{j} \ and \ \hat{k}, \hat{i} . \hat{i}=\hat{j} \cdot \hat{j}=\hat{k} \cdot \hat{k}=1\)
7. In the case of orthogonal unit vectors, \(\hat{i}, \hat{j} \ and \ \hat{k}, \hat{i}, \hat{j}=\hat{j}, \hat{k}=\hat{k} \cdot \hat{k}=0\)
8. Work is the example for dot product.