Math Problem Statement
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.
Solution
To find the magnitude and direction of the resultant vector , we must first determine the components of the resultant vector by summing the components of vectors and .
1. Components of the resultant vector:
2. Magnitude of the resultant vector:
The magnitude of the resultant vector is given by the Pythagorean theorem: Substituting the values:
3. Direction of the resultant vector:
The direction (angle ) 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: Substituting the values:
Final answer:
- Magnitude of the resultant vector:
- Direction of the resultant vector: (measured counterclockwise from the positive x-axis)
Would you like further details or have any questions?
Here are 5 related questions to further explore this topic:
- How do we calculate the components of a vector given its magnitude and direction?
- What happens if the vectors are in opposite directions?
- How can we calculate the resultant vector using graphical methods?
- How would the result change if one of the vectors were 3D (with a z-component)?
- What real-world applications use vector addition in physics or engineering?
Tip: Always break vectors into components before combining them—it simplifies calculations significantly!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Vector Addition
Trigonometry
Pythagorean Theorem
Formulas
Resultant vector components: R_x = a_x + b_x, R_y = a_y + b_y
Magnitude of the resultant vector: |R| = √(R_x^2 + R_y^2)
Direction of the resultant vector: θ = tan^(-1)(R_y / R_x)
Theorems
Pythagorean Theorem
Suitable Grade Level
Grades 10-12