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Show without expanding at any stage that: ` [0,sinalpha-cosalpha],[-sinalpha,0,sinbeta],[cosalphas-sinbeta,0]|=0 `

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Evaluate : Delta=|{:(0,sinalpha,-cosalpha),(-sinalpha,0,sinbeta),(cosalpha,-sinbeta,0):}| .

Show without expanding at any stage that: | (1,cosalpha-sinalpha, cosalpha+sinalpha),(1,cosbeta-sinbeta,cosbeta+sinbeta),(1, cosgamma-singamma,cosgamma+singamma)| =2 |(1,cosalpha, sinalpha),(1,cosbeta, sinbeta),(1,cosgamma,singamma)|

Knowledge Check

  • If A_(alpha)=[(cosalpha,-sinalpha),(sinalpha,cosalpha)] , then

    A
    `A_(alpha).A_((-alpha))=I`
    B
    `A_(alpha).A_((-alpha))=O`
    C
    `A_(alpha).A_(beta)=A_(alphabeta)`
    D
    `A_(alpha).A_(beta)=A_(alpha+beta)`

  • If cosalpha+cosbeta=0=sinalpha+sinbeta , then cos2alpha+cos2beta=?

    A
    `-2sin(alpha+beta)`
    B
    `2cos(alpha+beta)`
    C
    `2sin(alpha+beta)`
    D
    `-2cos(alpha+beta)`

  • Let A_(alpha)=[(cosalpha, -sinalpha,0),(sinalpha, cosalpha, 0),(0,0,1)] , then :

    A
    `A_(alpha+beta)=A_(alpha)A_(beta)`
    B
    `A_(alpha)^(-1)=A_(-alpha)`
    C
    `A_(alpha)^(-1)=-A_(alpha)`
    D
    `A_(alpha)^(2)=-I`

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    costheta-sintheta=cosalpha-sinalpha

    Show by vector method that sin(alpha-beta)=sinalphacosbeta-cosalpha sinbeta.

    Evaluate the following: |[cosalpha, sinalpha],[sinalpha, cosalpha]|

    If A=[(0,sin alpha, sinalpha sinbeta),(-sinalpha, 0, cosalpha cosbeta),(-sinalpha sinbeta, -cosalphacosbeta, 0)] then (A) |A| is independent of alpha and beta (B) A^-1 depends only on beta (C) A^-1 does not exist (D) none of these

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