Represent vectors using Cartesian components

A vector in the Cartesian plane is represented by an ordered pair $\langle a,b\rangle$ , where $a$ and $b$ are its signed horizontal and vertical components, so the vector equals $a\mathbf{i}+b\mathbf{j}$ . The components determine the vector’s magnitude $\sqrt{a^2+b^2}$ and direction, and can be found from a diagram or from magnitude and angle using trigonometric relationships; negative components indicate motion opposite an axis. This scope excludes three-dimensional vectors and more abstract vector-space representations.

Detailed Explanation: Represent vectors using Cartesian components

A vector in the Cartesian plane is written as an ordered pair:

\vec{v}=\langle a,b\rangle

$a$ is the horizontal component: positive right, negative left.
$b$ is the vertical component: positive up, negative down.

The same vector can also be written as

\vec{v}=a\mathbf{i}+b\mathbf{j}.

Example

A vector has magnitude $10$ and makes an angle of $30^\circ$ above the positive horizontal axis. Represent it using Cartesian components.

Since the angle is measured from the horizontal:

\text{horizontal component}=10\cos(30^\circ)

\text{vertical component}=10\sin(30^\circ)

Calculate each component:

10\cos(30^\circ)=10\left(\frac{\sqrt{3}}{2}\right)=5\sqrt{3}

10\sin(30^\circ)=10\left(\frac{1}{2}\right)=5

Therefore, the vector is

\boxed{\vec{v}=\langle 5\sqrt{3},5\rangle}

or, in unit-vector form,

\boxed{\vec{v}=5\sqrt{3}\mathbf{i}+5\mathbf{j}}.

Both components are positive because the vector points right and up. As a check, its magnitude is

\sqrt{(5\sqrt{3})^2+5^2} =\sqrt{75+25} =\sqrt{100} =10,

which matches the given magnitude.

Learn by doing: Represent vectors using Cartesian components

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Cartesian Grid - Vector Between Displayed Points With Coordinates (Angle)

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