6.3 Addition of Vectors

6.3 Addition of Vectors

(i) Triangle Law of Addition:

If two vectors $\underline{u}$ and $\underline{v}$ are represented by the two adjacent sides $\overrightarrow{AB}$ and $\overrightarrow{BC}$ of a triangle such that the terminal point of $\underline{u}$ coincides with the initial point of $\underline{v}$, then the third side $\overrightarrow{AC}$ of the triangle gives the vector sum $\underline{u} + \underline{v}$ that is:

$\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC} \Rightarrow \underline{u} + \underline{v} = \overrightarrow{AC}$

By head to tail rule

[Figure 6.9]

(ii) Parallelogram Law of Addition:

If two vectors $\underline{u}$ and $\underline{v}$ are represented by two adjacent sides $\overrightarrow{AB}$ and $\overrightarrow{AC}$ of a parallelogram as shown in the Figure 6.10, then diagonal $\overrightarrow{AD}$ gives the sum or resultant of $\overrightarrow{AB}$ and $\overrightarrow{AC}$.

$\overrightarrow{AD} = \overrightarrow{AB} + \overrightarrow{AC}$

$\overrightarrow{AD} = \underline{u} + \underline{v}$

[Figure 6.10]

Example 6

If $\underline{a} = 2\underline{i} + 3\underline{j}, \underline{b} = 4\underline{i} - 6\underline{j}$, then find $\underline{a} + \underline{b}$.

Solution $\underline{a} + \underline{b} = (2\underline{i} + 3\underline{j}) + (4\underline{i} - 6\underline{j})$

$= (2 + 4)\underline{i} + (3 - 6)\underline{j}$

$= 6\underline{i} - 3\underline{j}$

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Skilled Practice!
If $\underline{a} = 10\underline{i} - \dfrac{1}{2}\underline{j}, \underline{b} = \dfrac{3}{4}\underline{i} - \dfrac{7}{6}\underline{j}$, then find $\underline{a} + \underline{b}$.