\(\overrightarrow { A } =4\hat { i } -2\hat { j } +\hat { k } \)
\(\overrightarrow { B } =5\hat { i } +3\hat { j } -4\hat { k } \)
Resultant vector = \(\overrightarrow { A } +\overrightarrow { B } \)
\(=\left| \begin{matrix} \hat { i } & \hat { j } & \hat { k } \\ 4 & -2 & 1 \\ 5 & 3 & -4 \end{matrix} \right| \)
\(=\hat{i}(8-3)+\hat{j}[5-(-16)]+\hat{k}(12+10)\)
Resultant vector = 5\(\hat { i } \) +21\(\hat { j } \) + 22\(\hat { k } \)

\(l = lm, \theta = 30°\)
\(l_n={ l }_{cos \theta }=1 \times cos \ 30=\frac { \sqrt { 3 } }{ 2 } m\)
\({ l }_{ y }=l_{sin \theta} = 1\times sin \ 30=\frac { 1 }{ 2 }=0.5m \)

(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.

\(\overrightarrow { { v }_{ 1 } } =5\hat { i } +6\hat { j } \)
\(\overrightarrow { { v }_{ 2 } } =2\hat { i } +3\hat { j } \)
\(\therefore \Delta \vec{r} =\vec{r}_{2}-\vec{r}_{1}=2 \hat{i}+3 \hat{j}-5 \hat{i}-6 \hat{j}  \)
\(=-3 \hat{i}-3 \hat{j}  \)
\(\Delta t =5 \mathrm{sec}  \)
\(\therefore \Delta v_{\text {avg }} s =\frac{\Delta \vec{r}}{\Delta t}  \)
\(=\frac{-3}{5}(\hat{i}+\hat{j})\)

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.