Home
Class 12
MATHS
If veca,vecb,vecc are mutually perpendic...

If `veca,vecb,vecc` are mutually perpendicular vectors of equal magnitude show that `veca+vecb+vecc` is equally inclined to `veca, vecb and vecc`

Text Solution

Verified by Experts

Since `veca,vecb and vecc` are mutually perpendicular vectrors, we have `veca.vecb=vecb.vecc=vecc.veca=0`
it is given that
`|veca|=|vecb|=|vecc|`
Let vector `veca+vecb+vecc` be inclined to `veca,vecb and vecc` at angle `theta_(1),theta_(2)and theta_(3)` respectively. then , we have
`costheta_(1)=((veca+vecb+vecc).veca)/(|veca+vecb+vecc||veca|)=(veca.veca+vecb.veca+vecc.veca)/(|veca+vecb+vecc||veca|)`
`(|veca|^(2))/(|veca+vecb+vecc||veca|)[vecb.veca.vecc.veca=0]`
`=(|veca|)/(|veca+vecb+vecc|)`
`costheta_(2)=((veca+vecb+vecc).vecb)/(|veca+vecb+vecc||vecb|)=(veca.veca+vecb.veca+vecc.vecb)/(|veca+vecb+vecc|.|vecb|)`
`=(|vecb|^(2))/(|veca+vecb+vecc|.|vecb|)[veca.vecb=vecc.vecc.vecb=0]`
`(|vecb|)/(|veca+vecb+vecc|)`
`costheta_(3)=((veca+vecb+vecc).vecc)/(|veca+vecb+vecc||vecc|)=(veca.vecc+vecb.vecc+vecc.vecc)/(|veca+vecb+vecc|.|vecc|)`
`(|vecc|^(2))/(|veca+vecb+vecc|.|vecc|)[veca.vecc=vecb.vecc=0]`
`= (|vecc|)/(|veca+vecb+vecc|)`
now as `|veca|=|vecb|=|vecc|, costheta_(1)=costheta_(2)=costheta_(3)`
`theta_(1)=theta_(2)=theta_(3)`
Hence, the vector `(veca+vecb+vecc)` is equally inclined to `veca,vecb and vecc`.
Promotional Banner

Topper's Solved these Questions

  • DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS

    CENGAGE ENGLISH|Exercise Exercise 2.1|18 Videos

  • DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS

    CENGAGE ENGLISH|Exercise Exercise 2.2|15 Videos

  • DETERMINANTS

    CENGAGE ENGLISH|Exercise All Questions|264 Videos

  • DIFFERENTIAL EQUATIONS

    CENGAGE ENGLISH|Exercise Matrix Match Type|5 Videos

Similar Questions

Explore conceptually related problems

if veca,vecb and vecc are mutally perpendicular vectors of equal magnitudes, then find the angle between vectors and veca+ vecb+vecc .

If veca,vecb,vecc are three mutually perpendicular vectors, then the vector which is equally inclined to these vectors is (A) veca+vecb+vecc (B) veca/|veca|+vecb/|vecb|+vec/|vecc| (C) veca/|veca|^2+vecb/|vecb|^2+vecc/|vecc|^2 (D) |veca|veca-|vecb|vecb+|vecc|vecc

If veca,vecb and vecc are three mutually perpendicular unit vectors and vecd is a unit vector which makes equal angle with veca,vecb and vecc , then find the value of |veca+vecb+vecc+vecd|^(2) .

If veca , vecb , vecc are three mutually perpendicular unit vectors, then prove that ∣ ​ veca + vecb + vecc | ​ = sqrt3 ​

Let veca vecb and vecc be pairwise mutually perpendicular vectors, such that |veca|=1, |vecb|=2, |vecc| = 2 , the find the length of veca +vecb + vecc .

Let veca vecb and vecc be pairwise mutually perpendicular vectors, such that |veca|=1, |vecb|=2, |vecc| = 2 , the find the length of veca +vecb + vecc .

Statement 1: veca, vecb and vecc arwe three mutually perpendicular unit vectors and vecd is a vector such that veca, vecb, vecc and vecd are non- coplanar. If [vecd vecb vecc] = [vecdvecavecb] = [vecdvecc veca] = 1, " then " vecd= veca+vecb+vecc Statement 2: [vecd vecb vecc] = [vecd veca vecb] = [vecdveccveca] Rightarrow vecd is equally inclined to veca, vecb and vecc .

if veca, vecb and vecc are there mutually perpendicular unit vectors and veca ia a unit vector then find the value of |2veca+ vecb + vecc |^2

If veca, vecb, vecc are any three non coplanar vectors, then [(veca+vecb+vecc, veca-vecc, veca-vecb)] is equal to

If veca, vecb, vecc are the unit vectors such that veca + 2vecb + 2vecc=0 , then |veca xx vecc| is equal to: