Forces as vectors

In a nutshell

Forces can be described using vectors which can be written in component form. Vectors describe both magnitude and direction, and can be resolved into components labelled $$\bf i$$ and $${\bf j}$$.  $$\bf i$$ and $${\bf j}$$ are always perpendicular.

$$\bf{i,j}$$ notation

Vectors describe the magnitude and direction of a force. A vector can be broken down into components labelled $${\bf i}$$ and $${\bf j}$$. A resultant force is equal to the vector sum of $$\bf{i}$$ and $$\bf{j}$$​ vectors. 

Procedure

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Example 1

The forces $$(2{\bf i} + 6{\bf j})$$, $$(-4{\bf i}+3{\bf j})$$​ and $$(x{\bf i} + y{\bf j})$$ act upon an object which has a resultant force of zero. What are the values of $$x$$ and $$y$$?

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Separate into $${\bf i}$$ and $${\bf j}$$ vectors.

$$(2 -4 +x){\bf i} + (6+3+y){\bf j}$$

There resultant force is zero, so the sum of vectors is equal to $$0$$.

​$$(2 -4 +x){\bf i} + (6+3+y){\bf j} =0$$

Solve for $$x$$ and $$y$$.

​$$\underline{xi+yj =2i-9j}$$

Magnitude and direction

​The diagram shows the vector $$\textbf{v}$$​ inclined at an angle $$\theta$$​ to the positive $$x$$​-axis.

​The magnitude of a vector is its length. You can use Pythagoras' theorem to calculate the magnitude of vector $$\textbf{v}$$​, written as $$|\textbf{v}|$$​.

The direction of vector is the angle $$\theta$$. It can be found using trigonometry. 

​$$\tan \theta=\dfrac{b}a$$​​

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Example 2

A vector is given as $${\bf a}= 4{\bf i}+3{\bf j}$$​. What is the magnitude and the direction of $$\bf a$$?

Find the magnitude of the vector.

$$\begin{aligned}{\bf a} &= 4{\bf i} + 3{\bf j}\\|{\bf a}| &= \sqrt{4^2 +3^2}\\|{\bf a}| &= \sqrt{16 +9}\\|{\bf a}| &= \sqrt{25}\\|{\bf a}| &=\underline{ 5}\\\end{aligned}$$​​

Find the value of $$\theta$$.

$$\begin{aligned}\tan \theta &= \dfrac34\\\theta &=\tan^{-1} \dfrac34\\\theta &=\underline{36.9\degree}\\\end{aligned}$$​​

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