Resultant velocity of a quadcopter drone

Expert intro: This vector question is about converting directional information from a diagram into component form. The usual method is to resolve each vector into horizontal and vertical parts using trigonometry, then write the result with unit vectors $\hat{i}$ and $\hat{j}$.

Detailed walkthrough

To express a vector in rectangular form, write it as

$\vec{v}=a\hat{i}+b\hat{j}$

where:

• $a$ is the horizontal component,
• $b$ is the vertical component.

For any vector shown in the diagram, use the angle and magnitude to resolve components:

• if the vector makes an angle $\theta$ with the positive $x$-axis, then

$\vec{v}=(|\vec{v}|\cos\theta)\hat{i}+(|\vec{v}|\sin\theta)\hat{j}$

• if the direction is measured from a different axis or from a negative direction, adjust the signs accordingly.

So each vector in the diagram should be written by calculating its horizontal and vertical parts separately and then combining them into $\hat{i}$ and $\hat{j}$ form.

If you provide the exact magnitudes and directions from the diagram, each vector can be written explicitly in rectangular form.

💡 Pitfall guide

Do not confuse the magnitude of a vector with its components. The horizontal and vertical parts usually have different values, and the signs matter. Another common mistake is using sine and cosine in the wrong positions when the angle is measured from a different axis.

🔄 Real-world variant

If a vector is given as a resultant of two perpendicular motions, the rectangular form is often the fastest way to add them. You can add all $\hat{i}$ components together and all $\hat{j}$ components together to find the resultant velocity.

🔍 Related terms

rectangular form, unit vector, vector components