Component Form and Magnitude
In this discussion vectors will be denoted by lowercase boldface variables. Examples of vector quantities are velocity, acceleration and force.
Figure 1 shows two vectors v and u as directed line segments.
The vector v is represented by the directed line segment $\stackrel{\rightharpoonup}{RS}$ and has an initial point at R and a terminal point at S.
Vectors are often represented in component form. Vector v in Figure 1 has an initial point at the origin (0,0) and is said to be in standard position. A vector in standard position can be represented by the coordinates of its terminal point. Thus vin component form = $\langle {v}_{1},{v}_{2}\rangle $.
The magnitude of v or $\stackrel{\rightharpoonup}{RS}$ is represented by $\parallel \stackrel{\rightharpoonup}{RS}\parallel $ or $\parallel v\parallel $ and is calculated using the Distance Formula, $\parallel v=\sqrt{{v}_{1}^{2}+{v}_{2}^{2}}\parallel $.
MAGNITUDE OF A VECTOR:
$\parallel v=\sqrt{{v}_{1}^{2}+{v}_{2}^{2}}\parallel $
Vector u represented by $\stackrel{\rightharpoonup}{GH}$ in Figure 1 is not in standard position as its initial point is not the origin (0,0). The component form and magnitude of vector u can be calculated as follows:
COMPONENT FORM OF DIRECTED LINE SEGMENT:
Initial point: G (g_{1}, g_{2})
Terminal Point: H (h_{1}, h_{2})
$\stackrel{\rightharpoonup}{GH}=\langle {h}_{1}{g}_{1},{h}_{2}{g}_{2}\rangle =\langle {u}_{1},{u}_{2}\rangle =u$
MAGNITUDE OF DIRECTED LINE SEGMENT:
Initial point: G (g_{1}, g_{2})
Terminal Point: H (h_{1}, h_{2})
$\stackrel{\rightharpoonup}{\parallel GH\parallel}=\sqrt{{\left({h}_{1}{g}_{1}\right)}^{2}+{\left({h}_{2}{g}_{2}\right)}^{2}}=\sqrt{{u}_{1}^{2}+{u}_{2}^{2}}$
Let's look at a couple examples.
Step 1: Identify the initial and terminal coordinates of the vector. 
Initial Point G: (2, 2) Terminal Point H: (4, 4) 

Step 2: Calculate the components of the vector. Subtract the xcomponent of the terminal point from the xcomponent of the initial point for your xcomponent of the vector. Do the same for the ycomponents. 
$u=\langle {h}_{1}{g}_{1},{h}_{2}{g}_{2}\rangle =\langle {u}_{1},{u}_{2}\rangle $ $u=\langle 4\left(2\right),42\rangle $ $\langle 2,2\rangle =u$ 

Step 3: Calculate the magnitude of the vector. 
u = $\sqrt{{u}_{1}^{2}+{u}_{2}^{2}}$ u = $\sqrt{{\left(2\right)}^{2}+{\left(2\right)}^{2}}$ $\sqrt{4+4}$ $\sqrt{8}$ 
Step 1: Identify the initial and terminal coordinates of the vector. Because vector v is in standard position it's initial point is (0,0) 
Initial Point: (0, 0) Terminal Point: (8, 2) 
Step 2: Calculate the components of the vector. Subtract the xcomponent of the terminal point from the xcomponent of the initial point for your xcomponent of the vector. Do the same for the ycomponents. In this case the vector is in standard form therefore the components of the vector are the same as the components of the terminal point. 
$v=\langle {v}_{1}0,{v}_{2}0\rangle =\langle {v}_{1},{v}_{2}\rangle $ $v=\langle 80,20\rangle $ $\langle 8,2\rangle =v$ 
Step 3: Calculate the magnitude of the vector. 
v = $\sqrt{{v}_{1}^{2}+{v}_{2}^{2}}$ v = $\sqrt{{\left(8\right)}^{2}+{\left(2\right)}^{2}}$ $\sqrt{64+4}$ $\sqrt{68}$ $2\sqrt{17}$ 
Related Links: Math algebra Scalar Multiplication and Vector Addition Unit Vectors Pre Calculus 
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