Vector Calculus Basics

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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?

v= (x₂-x₁)i + (y₂-y₁)j Signup and view all the answers

What is the formula for adding vectors?

(a₁+a₂)i + (b₁+b₂)j Signup and view all the answers

What is the formula for subtracting vectors?

(a₁-a₂)i + (b₁-b₂)j Signup and view all the answers

How do you find a unit vector with the same direction as vector v?

First find the magnitude ‖v‖= √(x)²+(y)², then plug into: v/‖v‖. Signup and view all the answers

How do you write vector v in terms of i and j given its magnitude and directional angle?

v= ‖v‖cosθi + ‖v‖sinθj Signup and view all the answers

What formula do you use to find the specified scalar?

= a₁(a₂) + b₁(b₂) Signup and view all the answers

What is the formula to find the angle between two vectors?

θ= cos⁻¹(v x w/‖v‖‖w‖) Signup and view all the answers

How can you determine if two vectors are parallel?

Find the angle between the vectors; if θ is 0 or 180 degrees, then the vectors are parallel. Signup and view all the answers

How do you determine if two vectors are orthogonal?

Find the dot product of the two vectors; if it equals 0, then they are orthogonal. Signup and view all the answers

What is the process to find progw(v) and decompose vector v into two components?

<ol> <li>Find progw(v) using: (v x w/‖w‖²)w, where progw(v) = v₁; 2. Find v₂ = v - v₁.</li> </ol> Signup and view all the answers

How do you find work in physics?

Use the formula: w= F(AB) or w= F(‖AB‖)cosθ. Signup and view all the answers

What steps do you follow to find work given two points?

<ol> <li>Find F in terms of i and j; 2. Find vector AB using v= (x₂-x₁)i + (y₂-y₁)j; 3. Find w by calculating the dot product of F and AB.</li> </ol> Signup and view all the answers

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:
1. Calculate magnitude of v: ‖v‖ = √(x)² + (y)²
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:
1. Compute the projection of v onto w using: progw(v) = (v · w/‖w‖²)w
2. Let v₁ = progw(v)
3. 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:
1. Derive the force F in terms of i and j using vector representation.
2. Calculate the vector AB: v = (x₂ - x₁)i + (y₂ - y₁)j
3. 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.