In physics velocity, displacement, force and acceleration are defined as vector quantities.

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.

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 | )

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

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.

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

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

Two forces F1 and F2 are used to pull an object. The angle between between the two forces is θ. For what value of θ is the magnitude of the resultant force equal to √(|F1|^{2} + |F2|^{2})

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)

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