After landing on an unexplored Klingon planet, Spock tests for the direction of the magnetic field by firing a beam of electrons in various directions and by observing the following: Electrons moving upward feel a magnetic force in the NW direction; Electrons moving horizontally North are pushed down; Electrons moving horizontally South-East are pushed upward. He naturally concludes that the magnetic field at this landing site is in which direction?

Magnetic field is in south west direction.

Explanation:

Let us represent various direction by i, j, k. i representing east, j representing north and k representing vertically upward direction.

magnetic field is represented vectorially as follows

B = B₀ ( - i - j)

In the first case velocity of electron

v = v k

Force = q (v x B)

= - e [ vk x B₀ ( - i - j) ]

= evB₀ (j - i)

Direction of force is north - west.

In the second case velocity of electron

v = vj

Force = - e [ vj x B₀ ( - i - j) ]

= - evB₀ k

force is downward

In the third case, velocity of electron

v = v (-j + i)

Force = - e [ v (-j + i) x B₀ ( - i - j) ]

= 2 evB₀ k

Force is upward.