Free SAT II Physics - Vectors - Solutions

Solutions to the Sat Physics subject questions on vectors, with detailed explanations.



  1. Which of the following is represented by a vector?
    I) velocity    II) speed    III) displacement    IV) distance     V) force    VI) acceleration
    A) I only
    B) III and IV only
    C) I , III, V and VI
    D) III, V and VI only
    E) None
    Solution - Explanations
    In physics velocity, displacement, force and acceleration are defined as vector quantities.


  2. Which of the following is not represented by a vector?
    I) mass    II) electric field    III) magnetic field    IV) current     V) voltage    VI) work
    A)I , IV, V and VI
    B) II and III
    C) All
    D) I , IV and V only
    E) I and IV only
    Solution - Explanations
    In physics mass, current, voltage and work are defined as scalar quantities and therefore are NOT represented by vectors.


  3. A and B are vectors with angle θ between them such that 0 < θ < 90. If | A |, | B | and | A + B | are the magnitudes of vectors A, B and A + B respectively, which of the following is true?
    A) | A + B | > | A | + | B |
    B) | A + B | = | A | + | B |
    C) | A + B | = √(| A |2 + | B |2)
    D) | A + B | < √(| A |2 + | B |2)
    E) | A + B | < | A | + | B |
    Solution - Explanations
    Start with the following identity
    (A + B)(A + B) = (A + B)(A + B)
    use scalar product to rewrite the above as follows
    | A + B | 2 = | A | 2 + | B | 2 + 2 A B
    rewrite as
    | A + B | 2 = | A | 2 + | B | 2 + 2 |A| |B | cos (θ)
    Since 0 < θ <90, we can write
    0 < cos (θ) < 1
    multiply all terms of inequality by 2 |A| |B | to obtain
    0 < 2 |A| |B | cos (θ) < 2 |A| |B |
    add | A | 2 + | B | 2 to all terms of above inequality to obtain
    | A | 2 + | B | 2 < | A | 2 + | B | 2 + 2 |A| |B | cos (θ) < | A | 2 + | B | 2 + 2 |A| |B |
    the above inequality can now be written as
    | A | 2 + | B | 2 < | A + B | 2 < ( | A | + | B | ) 2
    all terms of the above inequality are positive, we can write the following inequality taking the square root as follows
    √[ | A | 2 + | B | 2 ] < | A + B | < ( | A | + | B | )


  4. Which of the following is a unit vector in the same direction as vector A = 3 i - 4 j, where i and j are the unit vector along the x and y axis respectively?
    A) (3/5) i + (4/5) j
    B) (9/5) i - (16/5) j
    C) (3/5) i - (4/5) j
    D) - (3/5) i + (4/5) j
    E) (3/25) i - (4/25) j
    Solution - Explanations
    The unit vector u in the same direction as vector A = 3 i - 4 j is given by
    u = A / | A | = ( 3 i - 4 j ) / √(3 2 + (-4) 2)
    = (3/5) i - (4/5)j


  5. If A and B are vectors, then A ( A B ) =
    A) | A | 3
    B) 0
    C) | A | 2
    D) 1
    E) | A |
    Solution - Explanations
    The cross product A B gives a vector perpendicular to both vectors A and B and therefore the scalar product between vectors A and A B , which is perpendicular to A, is equal to zero.


  6. Vector U has a magnitude of 3 and points Northward. Vector V has a magnitude of 7 and points Eastward. | U + V | is the magnitude of vector U + V . Which of the following is true?
    A) | U + V | > 10
    B) | U + V | = 10
    C) | U + V | = √58
    D) | U + V | = √10
    E) | U + V | = 4
    Solution - Explanations
    The x and y axes are directed Eastward and Northward respectively and therefore the components of U + V are given as follows
    U + V = 3j + 7 i
    magnitude is now calculated
    | U + V | = √(7 2 + 3 2) = √ 58


  7. Given vectors U = 2 i + 2j and V = 2i - 2j. What angle does the vector U - V make with the positive x - axis?
    A) 0
    B) 45
    C) -90
    D) 90
    E) -45
    Solution - Explanations
    U - V = 2 i + 2j - (2i - 2j) = 4j
    U - V is proportional to the unit vector j which makes 90 with the positive x-axis. Hence U - V makes 90 with the positive x-axis


  8. Two forces F1 and F2 are used to pull an object. The angle between the two forces is θ. For what value of θ is the magnitude of the resultant force equal to √(|F1|2 + |F2|2)
    A) 90
    B) 135
    C) 45
    D) 180
    E) 0
    Solution - Explanations
    Let R be the resultant and write
    R = F1 + F2
    use scalar product to write
    RR = (F1 + F2) (F1 + F2)
    expand the terms on the right
    |R|2 = |F1|2 + |F2|2 + |F1| |F2| cos (θ)
    For θ = 90, cos (θ) = 0
    and
    |R|2 = |F1|2 + |F2|2
    taking the square root gives
    |R| = √ (|F1|2 + |F2|2)


  9. Two forces F1 = 3i + bj and F2 = 9i + 12j act on the same object.(i and j are the unit vectors along the positive x-and y- axes). For what value of b will the magnitude of the resultant force be minimum?
    A) 0
    B) - 12
    C) 9
    D) - 10
    E) 4
    Solution - Explanations
    Let R be the resultant force
    R = F1 + F2 = 3i + bj + 9i + 12j = 12i + (b + 12)j
    calculate magnitude
    R = √( 144 + (b + 12)2 )
    the quantity under the radical is positive and therefore a value of b that minimizes 144 + (b + 12)2 will also minimize R the magnitude
    144 + (b + 12)2 is a quadratic expression and it has a minimum value at b = -12 (position of vertex)


  10. Find m so that the vectors A = 5 i - 10 j and B = 2 m i + (1 / 2) j are parallel.
    A) 5
    B) - 40
    C) 8
    D) - 1 / 8
    E) - 5
    Solution - Explanations
    For vectors A and B to be parallel, there must be a real K so that A = K B
    A = 5 i - 10 j = K ( B = 2 m i + (1 / 2) j )
    components are equal, hence
    5 = K (2 m)
    -10 = K (1 / 2)
    Solve the second equation fo K: K = -20
    substitute K by -20 in the equation 5 = K (2 m)
    5 = -20 (2 m)
    solve for m
    m = - 5 / 40 = - 1 / 8

Answers to the Above questions

  1. C
  2. A
  3. E
  4. C
  5. B
  6. C
  7. D
  8. A
  9. B
  10. D



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