Home
Class 12
MATHS
If vec An d vec B are two vectors and k...

If ` vec An d vec B` are two vectors and `k` any scalar quantity greater than zero, then prove that `| vec A+ vec B|^2lt=(1+k)| vec A|^2+(1+1/k)| vec B|^2dot`

Text Solution

Verified by Experts

We know, `(1+k)|A|^(2)(1+(1)/(k))|B|^(2)`
`=|A|^(2)+k|A|^(2)+|B|^(2)+(1)/(k)|B|^(2)` . . . (i)
Also, `k|A|^(2)+(1)/(k)|B|^(2)ge2(k|A|^(2)*(1)/(k)|B|^(2))^((1)/(2))=1|A|*|B|` . . . (ii)
(since, arithmetic mean `ge` Geometric mean)
So, `(1+k)|A|^(2)+(1+(1)/(k))|B|^(2) ge |A|^(2)+|B|^(2)+2|A|*|B|`
`=(|A|+|B|)^(2)` [using eqs. (i) and (ii)]
And also, `|A|+|B| ge |A+B|`
hence, `(1+k)|A|^(2)+(1+(1)/(k))|B|^(2)ge|A+B|^(2)`.
Promotional Banner

Topper's Solved these Questions

  • VECTOR ALGEBRA

    ARIHANT MATHS|Exercise Exercise For Session 1|7 Videos

  • VECTOR ALGEBRA

    ARIHANT MATHS|Exercise Exercise For Session 2|17 Videos

  • TRIGONOMETRIC FUNCTIONS AND IDENTITIES

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|19 Videos

Similar Questions

Explore conceptually related problems

If vec A nd vec B are two vectors and k any scalar quantity greater than zero,then prove that |vec A+vec B|^(2)<=(1+k)|vec A|^(2)+(1+(1)/(k))|vec B|^(2)

For any two vectors vec a and vec b prove that | vec a + vec b | <= | vec a | + | vec b |

For any to vectors vec A and vec B , prove that |vec A xx vec B|^2 = A^2 B^2 - ( vec A. vec B)^2 .

If vec a and vec b are two given vectors and k is any scalar,then find the vector vec r satisfying vec r xxvec a+kvec r=vec b

For any vector vec a and vec b prove that |vec a+vec b|<=|vec a|+|vec b|

If vec a and vec b are two vectors such that |vec a+vec b|=|vec a|, then prove that vector 2vec a+vec b is perpendicular to vector vec b

For any two vectors vec a, vec b | vec a * vec b | <= | vec a || vec b |

If vec a,vec b are two vectors such that |vec a+vec b|=|vec b|, then prove that vec a+2vec b is perpendicular to vec a .

If vec a,vec b are two vectors such that |vec a+vec b|=|vec b|, then prove that vec a+2vec b is perpendicular to vec a .

If vec a,vec b, are two vectors such that |vec a+vec b|=|vec a|, then prove that 2vec a+vec b is perpendicular to vec b .