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IBMathsDPApplications & Interpretation (AI)HLFlashcards3. Geometry & TrigonometryModelling with Vectors

Modelling with Vectors (DP IB Applications & Interpretation (AI))

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  • FrontKinematics with Vectors

    An object is moving with velocity vector bold italic v, how do you find the speed of the object?

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  • An object is moving with velocity vector bold italic v, how do you find the speed of the object?

    An object is moving with velocity vector bold italic v.

    You can find the speed by finding the magnitude of the velocity vector, open vertical bar bold italic v close vertical bar.

  • How can you find the shortest distance between two moving objects?

    To find the shortest distance between two moving objects, find the time at which the magnitude of the displacement vector between the two objects is at a minimum:

    1. Write the position vector of both objects in terms of the time t.

    2. Find the displacement vector between the two objects in terms of t.

    3. Find the magnitude of the displacement vector and square it, d squared.

    4. Use calculus to find the value of t for which d squared is at its minimum.

    5. Substitute the value of t back into d squared and take the positive square root.

  • True or False?

    The direction of a vector in 2D must be given as a bearing.

    False.

    The direction of a vector in 2D can be given as a bearing, however, it is commonly given as an angle from the positive x-axis (the base vector bold italic i). It can also be given as an angle from the base vector bold italic j.

  • How can you find the direction of a vector in 2D?

    You can find the direction of a vector in 2D by:

    • sketching the vector,

    • forming a right-angled triangle using the vectors bold italic i and bold italic j,

    • using trigonometry to find an angle in the triangle,

    • and stating the angle and direction from either bold italic i or bold italic j.

  • An object is moving with constant velocity, bold italic v, along a straight line.

    The position vector of the starting point is bold italic r subscript bold 0.

    What is an equation for the position vector of the object after time t?

    For an object moving with constant velocity, bold italic v, along a straight line, the position of the object at the time, t, can be given by bold italic r equals bold italic r subscript 0 plus t bold italic v

    Where:

    • bold italic r subscript 0 is the position vector of the object when t equals 0

    • bold italic v is the vector for the constant velocity

    • t is the time

  • If the displacement of an object from the origin is given by the vector bold italic r open parentheses t close parentheses equals open parentheses table row cell f subscript 1 open parentheses t close parentheses end cell row cell f subscript 2 open parentheses t close parentheses end cell end table close parentheses, where t is the time, how would you find the vector for the velocity at time t?

    If the displacement of an object from the origin is given by the vector bold italic r open parentheses t close parentheses equals open parentheses table row cell f subscript 1 open parentheses t close parentheses end cell row cell f subscript 2 open parentheses t close parentheses end cell end table close parentheseswhere t is the time, then you would differentiate each component with respect to t to find the vector for the velocity at time t.

    bold italic v open parentheses t close parentheses equals fraction numerator d bold italic r over denominator d t end fraction equals open parentheses table row cell f subscript 1 apostrophe open parentheses t close parentheses end cell row cell f subscript 2 apostrophe open parentheses t close parentheses end cell end table close parentheses.

  • If the velocity of an object is given by the vector bold italic v open parentheses t close parentheses, where t is the time, how would you find the position vector of the object at time t?

    If the velocity of an object is given by the vector bold italic v open parentheses t close parentheses, where t is the time, then you can find the position vector of the object at time t by integrating the velocity vector.

    You need to remember to include a constant of integration for each component.

    bold italic r open parentheses t close parentheses equals integral bold italic v open parentheses t close parentheses d t plus open parentheses table row cell c subscript 1 end cell row cell c subscript 2 end cell end table close parentheses.

  • If the displacement of an object from the origin is given by the vector bold italic r open parentheses t close parentheses equals open parentheses table row cell f subscript 1 open parentheses t close parentheses end cell row cell f subscript 2 open parentheses t close parentheses end cell end table close parentheses, where t is the time, how would you find the vector for the acceleration at time t?

    If the displacement of an object from the origin is given by the vector bold italic r open parentheses t close parentheses equals open parentheses table row cell f subscript 1 open parentheses t close parentheses end cell row cell f subscript 2 open parentheses t close parentheses end cell end table close parentheseswhere t is the time, then you would differentiate each component with respect to t twice to find the vector for the acceleration at time t.

    bold italic a open parentheses t close parentheses equals fraction numerator d squared bold italic r over denominator d t squared end fraction equals open parentheses table row cell f subscript 1 apostrophe apostrophe open parentheses t close parentheses end cell row cell f subscript 2 apostrophe apostrophe open parentheses t close parentheses end cell end table close parentheses.

  • True or False?

    If bold italic a open parentheses t close parentheses equals open parentheses table row 3 row cell 2 t end cell end table close parentheses is the acceleration of an object at time t, the velocity at time t is bold italic v open parentheses t close parentheses equals open parentheses table row cell 3 t end cell row cell t squared end cell end table close parentheses.

    False.

    If bold italic a open parentheses t close parentheses equals open parentheses table row 3 row cell 2 t end cell end table close parentheses is the acceleration of an object at time t, the velocity at time t is bold italic v open parentheses t close parentheses equals open parentheses table row cell 3 t plus c subscript 1 end cell row cell t squared plus c subscript 2 end cell end table close parentheses.

    You need to know the velocity at a particular time in order to find the exact function for the velocity at time t.

  • True or False?

    If bold italic a open parentheses t close parentheses equals open parentheses table row 2 row cell 6 t end cell end table close parentheses is the acceleration of an object at time t, its position vector at time t is bold italic r open parentheses t close parentheses equals open parentheses table row cell t squared plus c subscript 1 plus d subscript 1 end cell row cell t cubed plus c subscript 2 plus d subscript 2 end cell end table close parentheses.

    False.

    If bold italic a open parentheses t close parentheses equals open parentheses table row 2 row cell 6 t end cell end table close parentheses is the acceleration of an object at time t, its position vector at time t is bold italic r open parentheses t close parentheses equals open parentheses table row cell t squared plus c subscript 1 t plus d subscript 1 end cell row cell t cubed plus c subscript 2 t plus d subscript 2 end cell end table close parentheses.

    When you integrate for a second time, you need to remember to integrate the first pair of constants of integration.

  • True or False?

    If bold italic v open parentheses t close parentheses equals open parentheses table row 2 row cell 6 t end cell end table close parentheses is the velocity of an object at time t, its acceleration at time t is bold italic a open parentheses t close parentheses equals open parentheses table row 0 row 6 end table close parentheses.

    True.

    If bold italic v open parentheses t close parentheses equals open parentheses table row 2 row cell 6 t end cell end table close parentheses is the velocity of an object at time t, its acceleration at time t is bold italic a open parentheses t close parentheses equals open parentheses table row 0 row 6 end table close parentheses.

    You differentiate the components of the velocity vector to find the acceleration vector.

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    1. Topic 1

      Number Toolkit

    2. Topic 2

      Exponentials & Logs

    3. Topic 3

      Sequences & Series

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      Financial Applications

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      Complex Numbers

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    7. Topic 7

      Matrices

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      Eigenvalues & Eigenvectors

    1. Topic 1

      Linear Functions & Graphs

    2. Topic 2

      Further Functions & Graphs

    3. Topic 3

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      Functions Toolkit

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      Transformations of Graphs

    6. Topic 6

      Modelling with Logarithmic, Logistic & Piecewise Functions

    1. Topic 1

      Geometry Toolkit

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      Voronoi Diagrams

    6. Topic 6

      Matrix Transformations

    7. Topic 7

      Vector Properties

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      Vector Equations of Lines

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      Modelling with Vectors

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      Graph Theory

    1. Topic 1

      Statistics Toolkit

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      Correlation & Regression

    3. Topic 3

      Non-linear Regression

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      Probability

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      Random Variables

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      Binomial Distribution

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      Normal Distribution

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      Combinations of Normal Distributions & Sample Mean Distributions

    10. Topic 10

      Poisson Distribution

    11. Topic 11

      Hypothesis Testing using the Chi-squared Distribution

    12. Topic 12

      Hypothesis Testing for Population Parameters

    13. Topic 13

      Transition Matrices & Markov Chains

    1. Topic 1

      Differentiation

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      Further Differentiation

    3. Topic 3

      Integration

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      Further Integration

    5. Topic 5

      Kinematics

    6. Topic 6

      Differential Equations

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      Coupled & Second Order Differential Equations