To express vector $\vec{A}$ in terms of the unit vectors $\hat{i}$ and $\hat{j}$, we need to break down the vector into its components along the x-axis and y-axis.

Step-by-Step Solution

Given:

• Magnitude of $\vec{A} = 3.60 \, \text{m}$
• Angle with the positive x-axis is $70^\circ$

1. Find the components of $\vec{A}$:

The vector $\vec{A}$ can be expressed in terms of its x (horizontal) and y (vertical) components:

$\vec{A} = A_x \hat{i} + A_y \hat{j}$

where:

• $A_x = |\vec{A}| \cos(\theta)$
• $A_y = |\vec{A}| \sin(\theta)$

2. Calculate $A_x$ and $A_y$:

Using the given magnitude and angle:

$A_x = 3.60 \, \text{m} \times \cos(70^\circ)$ $A_y = 3.60 \, \text{m} \times \sin(70^\circ)$

3. Compute the components using trigonometric values:

$\cos(70^\circ) \approx 0.342$ $\sin(70^\circ) \approx 0.940$

Substitute these values:

$A_x = 3.60 \times 0.342 = 1.2312 \, \text{m}$ $A_y = 3.60 \times 0.940 = 3.384 \, \text{m}$

4. Express $\vec{A}$ in terms of $\hat{i}$ and $\hat{j}$:

$\vec{A} = 1.2312 \, \text{m} \, \hat{i} + 3.384 \, \text{m} \, \hat{j}$

Final Answer

$\vec{A} = 1.23 \, \hat{i} + 3.38 \, \hat{j} \, \text{m}$

Would you like more details or have any questions about this solution?

Here are some related questions for further exploration:

1. How would the components of $\vec{B}$ be expressed in terms of $\hat{i}$ and $\hat{j}$?
2. How can you calculate the magnitude of a resultant vector given two vectors?
3. What would change if the angle was measured from the negative x-axis?
4. How do you determine the direction of a vector given its components?
5. Can you convert between polar and Cartesian coordinates for any vector?

Tip: Always use trigonometric identities to break down vectors into their components. It simplifies the calculation and allows for easy addition and subtraction of vectors.