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Questions and Answers
What is the definition of a unit vector?
What is the definition of a unit vector?
Given the vectors A(2, -7) and B(-1, 5), how would you express vector AB in terms of unit vectors?
Given the vectors A(2, -7) and B(-1, 5), how would you express vector AB in terms of unit vectors?
What is the unit vector in the same direction as v = (6, 3)?
What is the unit vector in the same direction as v = (6, 3)?
What comprises the vertical and horizontal components of a force of 630N at an angle of 70 degrees?
What comprises the vertical and horizontal components of a force of 630N at an angle of 70 degrees?
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Which statement correctly describes vectors AB and CD for A(3,5), B(6,9), C(-4,-4), D(-2,0)?
Which statement correctly describes vectors AB and CD for A(3,5), B(6,9), C(-4,-4), D(-2,0)?
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If the length of vector v is 15 and the angle Θ is 125 degrees, what is the ordered pair representing the directed distance of vector v?
If the length of vector v is 15 and the angle Θ is 125 degrees, what is the ordered pair representing the directed distance of vector v?
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What is the direction angle of the vector -5i + 9j?
What is the direction angle of the vector -5i + 9j?
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How do you express the resulting force and direction when one person kicks a soccer ball with 55N west and another with 70N south?
How do you express the resulting force and direction when one person kicks a soccer ball with 55N west and another with 70N south?
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Study Notes
Algebraic Vectors in 2- and 3-Space
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Vector Representation: Vectors can be represented algebraically. A vector AB in two dimensions is expressed as (x₂ - x₁, y₂ - y₁), where (x₁, y₁) and (x₂, y₂) are coordinate points.
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Unit Vectors: A unit vector is parallel to a coordinate axis (x, y, or z) and has a length of 1.
- In two dimensions: i = (1, 0) and j = (0, 1)
- In three dimensions: i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1)
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Vector Magnitude (Length): The length of a vector is calculated using the Pythagorean theorem.
Vector Components and Direction
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Component Form: A vector's component form uses its magnitude and direction angle to find its horizontal and vertical components.
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Direction Angle: Direction angles are used to find the angle of a vector from the horizontal/positive x-axis.
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Magnitude and Direction: A vector's magnitude (or length) and its direction angle fully describe the vector.
Unit Vectors
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Definitions: Unit vectors have length 1 and share the same direction as another vector.
- Finding unit vectors is about scaling a vector down to a unit length without changing its direction.
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Formula: A unit vector in the same direction as vector V is (V/|V|)
Problems (Examples from the notes)
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Force Components: Finding the horizontal and vertical components of a force vector with given magnitude and direction angle.
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Vector Addition: Calculating the resultant force (vector sum) when two or more forces act simultaneously.
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Equivalent Vectors: Determining if two vectors (e.g., AB and CD) are the same, meaning they have the same magnitude and direction.
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Determining unit vectors: Finding a unit vector with the same direction as a given vector V.
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Calculating directed distance: Finding the ordered pair of a vector given its magnitude and direction.
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Direction angle of a given vector: Finding the direction angle of a given vector, commonly expressed in degrees from the x-axis.
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Description
This quiz focuses on the representation, components, and properties of algebraic vectors in two and three dimensions. Explore concepts like unit vectors, vector magnitude, and direction angles to enhance your understanding of vector mathematics.
