4.3.1a Addition of Vectors (Examples)

Example 1:

Find

(a) The resultant vector of the addition of the two parallel vectors above

(b) The magnitude of the resultant vector.

Solution:

(a)

Resultant vector

= addition of the two vectors

= \vec{P Q} + \vec{R S}

(b)

Magnitude of the resultant vector

\begin{array}{l} = | \vec{P Q} | + | \vec{R S} | \\ = | \sqrt{6^{2} + 8^{2}} | + | \sqrt{6^{2} + 8^{2}} | \\ = 10 + 10 \\ = 20 units \end{array}

Example 2:

The diagram above shows a parallelogram OABC. M is the midpoint of BC. Vector

\vec{O A} = \underset{˜}{a} and \vec{O C} = \underset{˜}{c} .

Find the following vectors in terms of

\underset{˜}{a} and \underset{˜}{c} .

\begin{array}{l} (a) \vec{O B} \\ (b) \vec{M B} \\ (c) \vec{O M} \end{array}

Solution:

(a)

\begin{array}{l} \vec{O B} = \vec{O A} + \vec{A B} \leftarrow Triangle Law of addition \\ = \vec{O A} + \vec{O C} \leftarrow \vec{A B} = \vec{O C} \\ = \underset{˜}{a} + \underset{˜}{c} \end{array}

(b)

\begin{array}{l} \vec{M B} = \frac{1}{2} \vec{C B} \leftarrow M is the midpoint of C B \\ = \frac{1}{2} \vec{O A} \\ = \frac{1}{2} \underset{˜}{a} \end{array}

(c)

\begin{array}{l} \vec{O M} = \vec{O C} + \vec{C M} \leftarrow Triangle Law of addition \\ = \vec{O C} + \vec{M B} \leftarrow \vec{C M} = \vec{M B} \\ = \underset{˜}{c} + \frac{1}{2} \underset{˜}{a} \end{array}