Vectors - The Basics — Question 2

Question 2

A vector in the plane has magnitude $12$, points into Quadrant II, and makes an angle of $35^\circ$ with the negative $x$-axis. Write it in component form.

Question 2 – Solution

Identify the initial and terminal information first. Build the vector by subtracting coordinates, then use the magnitude formula or normalize only after the components are correct.

Its standard-position angle is $180^\circ-35^\circ=145^\circ$.

Use $\vec v=\|\vec v\|\langle\cos\theta,\sin\theta\rangle$.

Thus $\boxed{\vec v=12\langle-\cos35^\circ,\sin35^\circ\rangle}$. The signs agree with Quadrant II.

The result follows from the defining vector formulas used above, and each component, magnitude, or scalar condition has been checked against the information in the question.

A vector separates two ideas: its components describe displacement, while its magnitude and unit vector describe size and direction. Always check signs and confirm that a unit vector has magnitude $1$.