Skill: 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.