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Questions and Answers
What is the correct representation of a vector in MATLAB?
What is the correct representation of a vector in MATLAB?
Which operation results in a scalar quantity?
Which operation results in a scalar quantity?
What does the length of a unit vector equal?
What does the length of a unit vector equal?
Which equation represents vector subtraction?
Which equation represents vector subtraction?
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What is the result of performing a cross product on two vectors?
What is the result of performing a cross product on two vectors?
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What is the correct formula for calculating the cross product of vectors 𝑎 and 𝑏?
What is the correct formula for calculating the cross product of vectors 𝑎 and 𝑏?
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Which of the following correctly describes a Row Vector?
Which of the following correctly describes a Row Vector?
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In MATLAB, how is a Column Vector created?
In MATLAB, how is a Column Vector created?
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Which of the following statements about a Square Matrix is true?
Which of the following statements about a Square Matrix is true?
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What operation transforms a Row Vector into a Column Vector?
What operation transforms a Row Vector into a Column Vector?
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Study Notes
Vectors
- A scalar has magnitude but no direction
- A vector has both magnitude and direction
- Vectors can represent position, velocity, acceleration, force, torque, angular velocity, angular acceleration, linear momentum, angular momentum, heat flux, magnetic flux
- Vector a has three components: a = ax i + ay j + az k
- i, j, and k are unit vectors in the x, y, and z directions
- The length of a unit vector is one
- The length of vector a is called its magnitude: a = √(𝑎𝑥2 + 𝑎𝑦2 + 𝑎𝑧2)
- Vectors are represented in MATLAB as: a = [𝑎𝑥 , 𝑎𝑦 , 𝑎𝑧] or [𝑎𝑥 𝑎𝑦 𝑎𝑧]
- Vector addition adds corresponding components of two vectors: a + b = (𝑎𝑥 +𝑏𝑥 ), (𝑎𝑦 + 𝑏𝑦 ), (𝑎𝑧 + 𝑏𝑧 )
- Vector subtraction subtracts corresponding components of two vectors: a - b = (𝑎𝑥 −𝑏𝑥 ), (𝑎𝑦 − 𝑏𝑦 ), (𝑎𝑧 − 𝑏𝑧 )
- Scalar × Vector: Distribute the scalar to each component of the vector: ca = c * [𝑎𝑥 , 𝑎𝑦 , 𝑎𝑧]
- Vector Dot Product: Sum of the products of corresponding components of two vectors: a ∙ b = 𝑎𝑥 𝑏𝑥 + 𝑎𝑦 𝑏𝑦 + 𝑎𝑧 𝑏𝑧
- Vector Cross Product: Results in a vector perpendicular to the plane defined by the two vectors being crossed: a × b = ( 𝑎 𝑏 sin 𝜃)𝑛
- n is the unit vector normal to the plane
- a × b = 𝑎𝑦 𝑏𝑧 − 𝑎𝑧 𝑏𝑦 𝑖 − 𝑎𝑥 𝑏𝑧 − 𝑎𝑧 𝑏𝑥 𝑗 + 𝑎𝑥 𝑏𝑦 − 𝑎𝑦 𝑏𝑥 𝑘
Matrices
- A Matrix is a rectangular array of numbers arranged in rows and columns
- The individual numbers in a Matrix are called elements
- Arow-column: Example, A23 = number in the second row, third column
- In MATLAB, use a comma to separate columns and a semi-colon to separate rows
- Row Vector: A Matrix with one row, multiple columns
- Column Vector: A Matrix with one column, multiple rows
- In MATLAB, create a Column Vector by using Semi-Colons to separate Rows
- Square Matrix: Number of Rows = Number of Columns
- Matrix Transpose: Switches the rows and columns
- Matrix Addition: The size of the two Matrices must be the same
- Scalar × Matrix Multiplication: The Scalar is Distributed to all of the elements of the Matrix
- Matrix × Matrix Multiplication: Inner sizes must be the same; outer sizes can be different
- Element × Element Multiplication: Defined only for arrays that have the same size
- Element × Element Division: Defined only for arrays that have the same size
- Element × Element Exponentiation: Defined only for arrays that have the same size
- Vectorized Functions: Results in a vector when applied to a vector
- Array Addressing: Selecting specific elements, rows, or columns for calculations
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Description
Explore the essential concepts of vectors, including their definitions, components, and operations in both physics and MATLAB. This quiz covers vector addition, subtraction, and scalar multiplication, providing a comprehensive understanding of vector properties and applications.
