Home
Class 12
MATHS
Let vec A(t) = f1(t) hat i + f2(t) hat ...

Let `vec A(t) = f_1(t) hat i + f_2(t) hat j and vec B(t) = g(t)hat i+g_2(t) hat j,t in [0,1],f_1,f_2,g_1 g_2` are continuous functions. If `vec A(t) and vec B(t)` are non-zero vectors for all `t and vec A(0) = 2hat i + 3hat j,vec A(1) = 6hat i + 2hat j, vec B(0) = 3hat i + 2hat i and vec B(1) = 2hat i + 6hat j` Then,show that `vec A(t) and vec B(t)` are parallel for some `t`.

Text Solution

Verified by Experts

`vecA(t)` is parallel to `vecB(t)` for some `t in [0, 1]` if and only if `(f_1(t))/(g_1(t))= (f_2(t))/(g_2(t))` for some `t in [0, 1]`
or `f_1(t)*g_2(t) = f_2(t)g_1(t)` for some `t in [0, 1]`
Let `h(t) = f_1(t)*g_2(t) -f_2(t)*g_1(t)`
`" " h(0) = f_1(0)*g_2(0)- f_2(0)*g_1(0)`
`" " = 2xx 2-3 xx 3=-5 lt 0`
`" " h(1) = f_1(1)*g_2(1)-f_2(1)*g_1(1)`
`" " = 6xx 6-2 xx 2 = 32 gt 0`
Since h is a continuous function, and `h(0)*h(1) lt 0`, there are some `t in [0,1]` for which `h(t) =0`, i.e.,
`vecA(t) and vecB(t)` are parallel vectors for this `t`.
Promotional Banner

Topper's Solved these Questions

  • INTRODUCTION TO VECTORS

    CENGAGE PUBLICATION|Exercise FILL IN THE BLANKS|2 Videos

  • INTRODUCTION TO VECTORS

    CENGAGE PUBLICATION|Exercise TRUE OR FALSE|1 Videos

  • INTRODUCTION TO VECTORS

    CENGAGE PUBLICATION|Exercise INTEGER TYPE|8 Videos

  • INTEGRALS

    CENGAGE PUBLICATION|Exercise All Questions|762 Videos

  • INVERSE TRIGONOMETRIC FUNCTIONS

    CENGAGE PUBLICATION|Exercise All Questions|541 Videos

Similar Questions

Explore conceptually related problems

Let vec a= hat i+2 hat j+ hat k , vec b= hat i- hat j+ hat ka n d vec c= hat i+ hat j- hat kdot Then find [vec a vec b vec c]

if vec a = 2 hat i- 3 hat j+hat k and vec b = hat i+2 hat j- 3hat k then vec aXvec b is

If vec (a) = 2 hat (i) - hat (j ) and vec (b) = 3 hat (i) - 2 hat (j) + 4 hat (k) , then the value of vec (a) xx vec (b) is -

If vec a= 2 hat i- hat j+ hat k and vec b = - hat i+3 hat j+4 hat k ,then veca.vecb =

Let vec (a) = 2 hat (i) - 2 hat (j) + hat (k) , vec (b) = hat (j) - hat (k) and vec(c ) = - hat (i) + 3 hat (j) + 2 hat (k) be three given vectors .Find vec(a) xx vec(b)

If vec a=x hat i+y hat j+z hat k , vec b=y hat i+z hat j+x hat k and vec c=z hat i+x hat j+y hat k , then vec axx( vec bxx vec c) is

If vec (alpha ) = - hat (i) + 2 hat (j) + hat (k) , vec(b) = 3 hat(i) + hat (j) + 2 hat (k) and vec(c ) = 2 hat (i) + hat (j) + 3 hat (k) , find [ vec(c ) vec(a) vec(b)]

Let vec (a) = 2 hat (i) - 2 hat (j) + hat (k) , vec (b) = hat (j) - hat (k) and vec(c ) = - hat (i) + 3 hat (j) + 2 hat (k) be three given vectors .Find vec(c ) xx (-vec(a))

If vec(a) = hat (i) +hat (j) , vec (b) = hat(i) - hat(j) and vec(c ) = 5 hat (i) + 2 hat (j) + 3 hat(k) , find the value of [ vec(b)vec (c ) vec(a)]

Find the projection of vector (vec(b)+vec(c )) on vector vec(a) where vec(a) = hat (i) + 2 hat (j) + hat (k) , vec(b) = hat (i) + 3 hat (j) + hat (k) and vec c = hat (i) + hat (k)