https://file.aiquickdraw.com/mathbot/file/1724944951782f2hncdvq.jpg

Write the  vector A⃗ in 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 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.

Learn how to express vector A in terms of unit vectors i and j using its magnitude and angle. This problem covers vector components, trigonometric calculations, and unit vector representation in meters. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation