Math Problem Statement
Solution
Let's break down and solve each part of the problem from the image.
Vectors in the Same Direction:
-
Since both vectors are in the same direction, you can add them directly:
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\quad \text{and} \quad 8\text{ cm} = 80\text{ mm}$$ Since they are in opposite directions: $$120\text{ mm} - 80\text{ mm} = 40\text{ mm} \rightarrow \text{ (final direction)}$$
The two vectors are in opposite directions, so we subtract the magnitudes. Convert to the same unit first:
Vectors in Opposite Directions:
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These vectors are in opposite directions. We can simply subtract their magnitudes: The negative sign means the resultant vector points to the right, so the result is: -
The vectors are in opposite vertical directions:
Vectors in Perpendicular Directions:
For this, we use the Pythagorean theorem to find the resultant of the perpendicular vectors: \quad \text{and} \quad \text{Vector 2: } 8\text{ dam} \uparrow$$ Using the Pythagorean theorem: $$R^2 = (12^2 + 8^2) = 144 + 64 = 208$$ So, $$R = \sqrt{208} \approx 14.42\text{ dam}$$. You would then graph this with a horizontal vector of $$12\text{ dam} \rightarrow$$ and a vertical vector of $$8\text{ dam} \uparrow$$.
Would you like more details or further explanations on any of these problems?
Here are some related questions to explore:
- What is the significance of direction in vector addition?
- How can you determine the resultant vector when dealing with multiple directions?
- Why do you use the Pythagorean theorem for perpendicular vectors?
- How would vector subtraction differ if the vectors were in the same direction?
- How does changing the unit (e.g., from cm to mm) impact vector calculations?
Tip: Always ensure all vectors are in the same units before performing operations like addition or subtraction.
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Math Problem Analysis
Mathematical Concepts
Vector Addition
Vector Subtraction
Pythagorean Theorem
Formulas
Vector addition in the same direction: V₁ + V₂
Vector subtraction in opposite directions: V₁ - V₂
Pythagorean theorem for perpendicular vectors: X² + Y² = R²
Theorems
Pythagorean Theorem
Suitable Grade Level
Grade 10-12
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