RECTANGULAR COMPONENTS OF A VECTOR

We denote the unit vector along x-axis by i and along y-axis by j.

Let us consider a point P(x,y) in the Cartesian plane. We want to find the position vector OP in terms of unit vectors i and j.

Draw perpendicular from P on x-axis meeting the x-axis at point A. As we know that for any vector u:

\vec{u} = \frac{\vec{u}}{∣ \vec{u} ∣} or \vec{u} = ∣ \vec{u} ∣ \hat{u}

i.e., Every vector is equal to the product of magnitude with its unit vector. So,

\overrightarrow{OA} = |\overrightarrow{OA}| \vec{i} = x\vec{i}

\overrightarrow{AP} = |\overrightarrow{AP}| \vec{j} = y\vec{j}

Now,

\overrightarrow{OP} = \overrightarrow{OA} + \overrightarrow{AP} = x\vec{i} + y\vec{j}

In Cartesian plane a vector u is expressed as:

\vec{u} = x\vec{i} + y\vec{j}

The magnitude of vector u is defined as:

∣ \vec{u} ∣ = \sqrt{x^{2} + y^{2}}

Two vectors are said to be equal if and only if they have same x and y components. If $\vec{u} = x_1\vec{i} + y_1\vec{j}$

= $x_{1} i$

+ $y_{1} j$

and $\vec{v} = x_2\vec{i} + y_2\vec{j}$

= $x_{2} i$

+ $y_{2} j$

then $\vec{u} = \vec{v}$

= $v$

if $x_1 = x_2$ and $y_1 = y_2$ .

If $\vec{p} = x\vec{i} + y\vec{j}$

= $x i$

+ $y j$

then negative of $\vec{p}$

is:

-\vec{p} = -x\vec{i} - y\vec{j}

If $\vec{w} = x\vec{i} + y\vec{j}$

= $x i$

+ $y j$

then $\vec{w}$

is called a unit vector if:

∣ \vec{w} ∣ = 1 \Rightarrow \sqrt{x^{2} + y^{2}} = 1 or x^{2} + y^{2} = 1

Rectangular Components of a Vector