Q4 of 33 Page 178

Show that the area of the triangle contained between the vectors a and b is one half of the magnitude of a × b.

Consider two vectors (OK) and (OM) are making angle θ which each other as shown in following figure,



Now,


In ΔOMN we can write the equation,



⇒


We know that magnitude of cross product of vectors a and b is,



= (∵)


= ×OK×MN (∵ = 1)


= 2×Area of ΔOMK


∴ Area of ΔOMK =


Hence Proved.


More from this chapter

All 33 →
2

In the HCl molecule, the separation between the nuclei of the two atoms is about 1.27 Å (1 Å = 10-10 m). Find the approximate location of the CM of the molecule, given that a chlorine atom is about 35.5 times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus.

 3

A child sits stationary at one end of a long trolley moving uniformly with a speed V on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner, what is the speed of the CM of the (trolley + child) system?

 5

Show that a.(b × c) is equal in magnitude to the volume of the parallelepiped formed on the three vectors , a, b and c.

 6

Find the components along the x, y, z axes of the angular momentum l of a particle, whose position vector is r with components x, y, z and momentum is p with components px, py and pz. Show that if the particle moves only in the x-y plane the angular momentum has only a z-component.