Questions and Answers
What is the formula to find the magnitude given 2 points?
‖x‖= √(x₂-x₁)²+(y₂-y₁)²
How do you determine if vectors u and v are equal?
Find the slope of u and v and see if they are equal.
What formula is used to find the magnitude given vector v in terms of i and j?
‖x‖= √(x)²+(y)²
What is the equation to write vector v in terms of i and j given points P1 and P2?
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What is the formula for adding vectors?
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What is the formula for subtracting vectors?
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How do you find a unit vector with the same direction as vector v?
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How do you write vector v in terms of i and j given its magnitude and directional angle?
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What formula do you use to find the specified scalar?
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What is the formula to find the angle between two vectors?
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How can you determine if two vectors are parallel?
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How do you determine if two vectors are orthogonal?
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What is the process to find progw(v) and decompose vector v into two components?
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How do you find work in physics?
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What steps do you follow to find work given two points?
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Study Notes
Finding Magnitude
- Magnitude of a vector between two points (P1 and P2) is calculated using the distance formula: ‖x‖ = √((x₂-x₁)² + (y₂-y₁)²)
- If vector v is expressed in terms of unit vectors i and j, apply the Pythagorean theorem: ‖v‖ = √(x)² + (y)²
Vectors Comparison
- To determine if two vectors u and v are equal, calculate their slopes. If slopes are identical, then u = v.
Vector Representation
- Vector v can be expressed in terms of unit vectors i and j using the equation: v = (x₂ - x₁)i + (y₂ - y₁)j
Vector Operations
- To add vectors, combine their components: Resultant = (a₁ + a₂)i + (b₁ + b₂)j
- To subtract vectors, use: Resultant = (a₁ - a₂)i + (b₁ - b₂)j
Unit Vectors
- Finding a unit vector in the direction of vector v requires these steps:
- Calculate magnitude of v: ‖v‖ = √(x)² + (y)²
- Normalize v by dividing by its magnitude: v/‖v‖
Vector in Polar Form
- Express a vector v based on its magnitude and directional angle θ: v = ‖v‖cosθi + ‖v‖sinθj
Dot Product
- To find a specified scalar using the dot product, apply the formula: a · b = a₁(a₂) + b₁(b₂)
Angle Between Vectors
- The angle θ between two vectors v and w can be calculated using the formula: θ = cos⁻¹((v · w)/(‖v‖‖w‖))
Vector Parallelism and Orthogonality
- Two vectors are parallel if the angle θ calculated between them is 0° or 180°.
- Two vectors are orthogonal if their dot product equals zero.
Vector Decomposition
- To decompose vector v into components parallel and orthogonal to vector w:
- Compute the projection of v onto w using: progw(v) = (v · w/‖w‖²)w
- Let v₁ = progw(v)
- Calculate v₂ as the difference: v₂ = v - v₁
Work Calculation
- Work done is defined as: w = F(AB) or equivalently, w = F(‖AB‖)cosθ
- For work calculation between two points:
- Derive the force F in terms of i and j using vector representation.
- Calculate the vector AB: v = (x₂ - x₁)i + (y₂ - y₁)j
- Determine work as the dot product of F and AB: w = a₁(a₂) + b₁(b₂)
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Description
This quiz covers fundamental concepts in vector calculus, including finding the magnitude of vectors given two points, determining the equality of vectors, and expressing vectors in terms of their components. Test your understanding of these essential skills in vector mathematics.
