The main objective of this question is to **resolve** the given vector into its **component** and **determine** its **magnitude**.

This question uses the concept of **Vector resolution**. A **vector resolution** is the **breaking** of such a **single vector** into** several vectors** in various **directions** that **collectively generate** the same **effect** as a **single vector**. Component **vectors** are the **vectors** created following **splitting**.

## Expert Answer

We have to **resolve** the given **vectors** into its **component**.

By using the **sine rule**, we get:

\[ \space \frac{F_2}{sin \space 70} \space = \space \frac{(F_2)_u}{sin \space 45} \space = \frac{(F_2)_v}{sin 65 } \]

Now **calculating** $ F_2 $ in the **direction** of $ u $.

**So**:

\[ \space \frac{F_2}{sin \space 70 } \space = \space \frac{(F_2)_u}{sin \space 45} \]

\[ \space (F_2)_u \space = \space \frac{F_2 \space \times \space sin \space 45 } {sin \space 70} \]

By **putting** the **value** of $F_2$, we get:

\[ \space (F_2)_u \space = \space \frac{500 \space \times \space sin \space 45 } {sin \space 70} \]

By **simplifying**, we get:

\[ \space (F_2)_u \space = \space 376.24 \]

Now **resolving** in the $ v $ direction.

\[ \space \frac{F_2}{sin \space 70 } \space = \space \frac{(F_2)_v}{sin \space 65} \]

\[ \space (F_2)_v \space = \space \frac{F_2 \space \times \space sin \space 65 } {sin \space 70} \]

By **putting** the value of $F_2$, we get:

\[ \space (F_2)_v \space = \space \frac{500 \space \times \space sin \space 65 } {sin \space 70} \]

By **simplifying**, we** get**:

\[ \space (F_2)_u \space = \space 482.24 \space N \]

Now **magnitude** is **calculated** as:

\[ \space F_2 \space = \space \sqrt{(F_2)^2_u \space + \space (F_2)^2_v} \]

By p**utting values**, we get:

\[ \space = \space \sqrt {(376.24)^2 \space + \space (482.24)^2 } \]

\[ \space F_2 \space = \space 611.65 \space N \]

## Numerical Answer

The **magnitude** of $ F_2 $ **resolving** into **components** is:

\[ \space F_2 \space = \space 611.65 \space N \]

## Example

In the **above question**, if the **magnitude** of $ F_2 $ is $ 1000 \space N $, find the **magnitude** of $F_2$ after **resolving** into its **components** $u$ and $v$.

By using the **sine rule**, we get:

\[ \space \frac{F_2}{sin \space 70} \space = \space \frac{(F_2)_u}{sin \space 45} \space = \frac{(F_2)_v}{sin 65 } \]

Now **calculating** $ F_2 $ in the **direction** of $ u $.

**So**:

\[ \space \frac{F_2}{sin \space 70 } \space = \space \frac{(F_2)_u}{sin \space 45} \]

\[ \space (F_2)_u \space = \space \frac{F_2 \space \times \space sin \space 45 } {sin \space 70} \]

By **putting** the **value** of $F_2$, we get:

\[ \space (F_2)_u \space = \space \frac{1000 \space \times \space sin \space 45 } {sin \space 70} \]

By **simplifying**, we get:

\[ \space (F_2)_u \space = \space 752.48 \]

Now **resolving** in the $ v $ direction.

\[ \space \frac{F_2}{sin \space 70 } \space = \space \frac{(F_2)_v}{sin \space 65} \]

\[ \space (F_2)_v \space = \space \frac{F_2 \space \times \space sin \space 65 } {sin \space 70} \]

By **putting** the value of $F_2$, we get:

\[ \space (F_2)_v \space = \space \frac{1000 \space \times \space sin \space 65 } {sin \space 70} \]

By **simplifying**, we** get**:

\[ \space (F_2)_u \space = \space 964.47 \space N \]

Now **magnitude** is **calculated** as:

\[ \space F_2 \space = \space \sqrt{(F_2)^2_u \space + \space (F_2)^2_v} \]

By **p****utting values**, we get:

\[ \space = \space \sqrt {(752.48)^2 \space + \space (964.47)^2 } \]

\[ \space F_2 \space = \space 1223.28 \space N \]