Vectors Overview and Components
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Questions and Answers

What is a vector?

A quantity with both magnitude and direction.

Define an algebraic vector.

v = < a, b > where a and b are components of vector v.

What is a component of a vector?

One of two parts of a vector.

What is the horizontal component of a vector?

<p>The first component of a vector, representing the horizontal aspect or x-coordinate.</p> Signup and view all the answers

What is the vertical component of a vector?

<p>The second component of a vector, representing the vertical aspect or y-coordinate.</p> Signup and view all the answers

What is a scientific vector?

<p>v = ai + bj where i = and j =</p> Signup and view all the answers

Define the magnitude of a vector.

<p>The length of a vector, represented as ‖v‖.</p> Signup and view all the answers

What is a unit vector?

<p>A vector u for which ‖u‖ = 1.</p> Signup and view all the answers

What is a zero vector?

<p>The vector whose magnitude is 0 and assigned no direction.</p> Signup and view all the answers

What is a line segment?

<p>The segment of a line defined by two points.</p> Signup and view all the answers

What is a directed line segment?

<p>A line segment bounded by points P and Q, indicating direction.</p> Signup and view all the answers

Define a position vector.

<p>A vector whose initial point is at the origin.</p> Signup and view all the answers

What is a velocity vector?

<p>A vector representing the direction and speed of an object.</p> Signup and view all the answers

Define a force vector.

<p>A vector representing the direction and amount of force acting on an object.</p> Signup and view all the answers

What is a direction angle?

<p>The angle α between position vector v and the x-axis.</p> Signup and view all the answers

How can you find a vector from its direction and magnitude?

<p>v = ‖v‖(cos αi + sin αj)</p> Signup and view all the answers

What are scalars in relation to vectors?

<p>Any real numbers; quantities that have only magnitude.</p> Signup and view all the answers

What is a scalar multiple?

<p>The result of multiplying a scalar by a vector.</p> Signup and view all the answers

What is the addition of vectors?

<p>Let v = a₁i + b₁j and w = a₂i + b₂j: v + w = (a₁ + a₂)i + (b₁ + b₂)j.</p> Signup and view all the answers

How is the subtraction of vectors defined?

<p>Let v = a₁i + b₁j and w = a₂i + b₂j: v - w = (a₁ - a₂)i + (b₁ - b₂)j.</p> Signup and view all the answers

Define scalar multiplication of vectors.

<p>Let v = a₁i + b₁j: αv = (αa₁)i + (αb₁)j.</p> Signup and view all the answers

What is the unit vector in the direction of v?

<p>u = v / ‖v‖.</p> Signup and view all the answers

What is a dot product?

<p>The multiplication of two vectors that returns a scalar.</p> Signup and view all the answers

How do you find the angle between vectors?

<p>cos θ = (u * v)/(‖u‖ ‖v‖)</p> Signup and view all the answers

What does orthogonal mean in vector terms?

<p>Describes two vectors which meet at a right angle.</p> Signup and view all the answers

How do you determine if vectors are orthogonal?

<p>Two vectors v and w are orthogonal if and only if v * w = 0.</p> Signup and view all the answers

What is decomposition in vector terms?

<p>To derive two vectors v₁ and v₂ from vector v with respect to vector w.</p> Signup and view all the answers

Define vector projection.

<p>The vector v₁ which is parallel to w when decomposing vector v.</p> Signup and view all the answers

What is work in physics as it relates to vectors?

<p>W = (magnitude of force)(distance) = (‖F‖)(‖→AB‖).</p> Signup and view all the answers

Study Notes

Vectors Overview

  • Vectors have both magnitude (length) and direction.
  • Algebraic vector representation: v = < a, b >.
  • Components of a vector are its coordinates, indicating its position in space.

Components of Vectors

  • Horizontal component: First part of a vector representing the x-coordinate.
  • Vertical component: Second part of a vector representing the y-coordinate.

Scientific Representation

  • Scientific vector form: v = ai + bj, where i and j denote unit vectors along the axes.

Magnitude and Properties

  • Magnitude of vector v = < a, b > calculated as ‖v‖ = √(a² + b²).
  • Unit vector: A vector with a magnitude of 1.
  • Zero vector: A vector with zero magnitude, lacking direction.

Basic Vector Definitions

  • Directed line segment: Line segment from point P to Q, defining a geometric vector.
  • Position vector: Origin at (0,0) and terminal point at (a,b).

Motion and Forces

  • Velocity vector: Indicates speed and direction of an object.
  • Force vector: Represents force's direction and magnitude acting on an object.

Angular and Directional Concepts

  • Direction angle (α): Angle formed between the position vector and x-axis, ranging from 0° to 360°.

Constructing Vectors

  • To find a vector from its magnitude and angle: v = ‖v‖(cos αi + sin αj).
  • Scalars: Real numbers that represent quantities with only magnitude.

Vector Operations

  • Scalar multiple of a vector modifies its magnitude and potentially its direction.
  • Vector addition involves combining components: v + w = (a₁ + a₂)i + (b₁ + b₂)j.
  • Vector subtraction is done similarly: v - w = (a₁ - a₂)i + (b₁ - b₂)j.

Unit Vectors and Projections

  • Unit vector in the direction of v calculated as u = v / ‖v‖.
  • Decomposition of vector v relates to orthogonal projections onto another vector w.

Dot Product and Orthogonality

  • Dot product: v * w = a₁a₂ + b₁b₂, resulting in a scalar.
  • Two vectors are orthogonal if their dot product equals zero.

Vector Decomposition

  • To decompose vector v relative to vector w:
    • Parallel component: v₁ = ((v * w)/(‖w‖²))w.
    • Orthogonal component: v₂ = v - v₁.

Work Done by Forces

  • Work (W) is calculated as the product of force magnitude and distance: W = (‖F‖)(‖→AB‖).
  • If force is applied at an angle, the work done is W = F * →AB.

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Description

This quiz covers the fundamental concepts of vectors, including their definitions, components, and scientific representation. You'll explore key properties such as magnitude, unit vectors, and applications in motion and force. Test your understanding of the foundational elements of vector algebra.

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