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