To find the components of the vector $\vec{a}$ in the $xy$-plane, we can break it down using trigonometry. Given:

• The direction of $\vec{a}$ is $249^\circ$ counterclockwise from the positive $x$-axis.
• The magnitude of $\vec{a}$ is $9.4 \, \text{m}$.

Step 1: Understanding the angle

The vector's direction is measured counterclockwise from the positive $x$-axis, so the angle with respect to the positive $x$-axis is $249^\circ$.

To compute the components of $\vec{a}$, we will use the following formulas:

$a_x = |\vec{a}| \cos(\theta)$ $a_y = |\vec{a}| \sin(\theta)$ where:

• $|\vec{a}| = 9.4 \, \text{m}$ (the magnitude),
• $\theta = 249^\circ$ (the angle with respect to the positive $x$-axis).

Step 2: Calculating the $x$-component

$a_x = 9.4 \cos(249^\circ)$

Step 3: Calculating the $y$-component

$a_y = 9.4 \sin(249^\circ)$

Now let's calculate these values.

The components of vector $\vec{a}$ are:

(a) The $x$-component: $a_x = -3.37 \, \text{m}$

(b) The $y$-component: $a_y = -8.78 \, \text{m}$

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How would the components change if the magnitude of the vector was doubled?
2. What are the $x$- and $y$-components if the angle was $90^\circ$ instead of $249^\circ$?
3. Can you express the direction and magnitude of a vector given only its components?
4. What is the resultant vector if you add another vector with a direction of $120^\circ$ and a magnitude of $5 \, \text{m}$?
5. How does the vector's angle affect the sign of its components?

Tip: Always ensure that the angle used is measured counterclockwise from the positive $x$-axis unless specified otherwise.