The vector a, has components ax = 60 cm and ay = 20 cm. The vector b has components bx = 30 cm and by = 70 cm. Determine the magnitude and direction of the resultant.

To find the magnitude and direction of the resultant vector \(\mathbf{R}\), we must first determine the components of the resultant vector by summing the components of vectors \(\mathbf{a}\) and \(\mathbf{b}\).

### 1. Components of the resultant vector:
\[
R_x = a_x + b_x = 60 \, \text{cm} + 30 \, \text{cm} = 90 \, \text{cm}
\]
\[
R_y = a_y + b_y = 20 \, \text{cm} + 70 \, \text{cm} = 90 \, \text{cm}
\]

### 2. Magnitude of the resultant vector:
The magnitude \(|\mathbf{R}|\) of the resultant vector is given by the Pythagorean theorem:
\[
|\mathbf{R}| = \sqrt{R_x^2 + R_y^2}
\]
Substituting the values:
\[
|\mathbf{R}| = \sqrt{(90)^2 + (90)^2} = \sqrt{8100 + 8100} = \sqrt{16200} \approx 127.28 \, \text{cm}
\]

### 3. Direction of the resultant vector:
The direction (angle \(\theta\)) of the resultant vector relative to the x-axis is found using the arctangent of the ratio of the y-component to the x-component:
\[
\theta = \tan^{-1} \left( \frac{R_y}{R_x} \right)
\]
Substituting the values:
\[
\theta = \tan^{-1} \left( \frac{90}{90} \right) = \tan^{-1}(1) = 45^\circ
\]

### Final answer:
- Magnitude of the resultant vector: \(127.28 \, \text{cm}\)
- Direction of the resultant vector: \(45^\circ\) (measured counterclockwise from the positive x-axis)

Would you like further details or have any questions?

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Here are 5 related questions to further explore this topic:
1. How do we calculate the components of a vector given its magnitude and direction?
2. What happens if the vectors are in opposite directions?
3. How can we calculate the resultant vector using graphical methods?
4. How would the result change if one of the vectors were 3D (with a z-component)?
5. What real-world applications use vector addition in physics or engineering?

**Tip**: Always break vectors into components before combining them—it simplifies calculations significantly!

Learn how to find the magnitude and direction of a resultant vector with given components. This problem demonstrates vector addition, trigonometry, and the Pythagorean theorem. Suitable for students in Grades 10-12.

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