Write the  vector  A⃗  in  Two vectors are shown in an x y plane. Both vectors start at point O located at the origin. Vector 'A' is located in the first quadrant, makes an angle of 70.0 degrees with the positive x axis, and has a length of 3.60 meters. Vector B is located in the third quadrant, makes an angle of 30.0 degrees with the negative x axis, and has a length of 2.40 meters. in terms of the unit vectors  i^  and  j^ .
Express your answer in terms of the unit vectors  i^  and  j^ . Use the 'unit vector' button to denote unit vectors in your answer. Express the coefficients in meters.

To express the vector \(\vec{A}\) in terms of the unit vectors \(\hat{i}\) and \(\hat{j}\), we can use the following steps:

1. **Determine the components of vector \(\vec{A}\):**

   - The magnitude of \(\vec{A}\) is \(3.60\) meters.
   - The angle \(\theta_A\) that \(\vec{A}\) makes with the positive x-axis is \(70.0^\circ\).

2. **Calculate the components of \(\vec{A}\):**

   - The x-component (\(A_x\)) is given by:
     \[
     A_x = A \cos(\theta_A) = 3.60 \cos(70.0^\circ)
     \]
   - The y-component (\(A_y\)) is given by:
     \[
     A_y = A \sin(\theta_A) = 3.60 \sin(70.0^\circ)
     \]

3. **Express \(\vec{A}\) in unit vector notation:**
   \[
   \vec{A} = A_x \hat{i} + A_y \hat{j}
   \]

Now let's compute the components:

\[
A_x = 3.60 \cos(70.0^\circ) \approx 1.23 \, \text{meters}
\]
\[
A_y = 3.60 \sin(70.0^\circ) \approx 3.38 \, \text{meters}
\]

Thus, the vector \(\vec{A}\) in unit vector notation is:

\[
\vec{A} \approx 1.23 \hat{i} + 3.38 \hat{j} \, \text{meters}
\]

### Do you want details or have any questions?
Here are 5 related questions:
1. How do you find the unit vector of a given vector?
2. What is the significance of the angle between the vector and the x-axis?
3. How would the components change if vector \(\vec{A}\) was in the second quadrant?
4. How can you determine the direction of a vector using its components?
5. What role does trigonometry play in vector decomposition?

**Tip:** Always ensure your calculator is set to the correct mode (degrees or radians) when calculating trigonometric functions.

Learn how to express vector A in terms of unit vectors i and j in an x y plane. This problem involves calculating vector components using trigonometry and is suitable for high school students studying vectors and trigonometry.

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