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If veca , vecb , vecc and vecd are four ...

If `veca , vecb , vecc and vecd` are four non-coplanar unit vectors such that `vecd` makes equal angles with all the three vectors `veca, vecb, vecc` then prove that `[vecd vecavecb]=[vecd veccvecb]=[vecd veccveca]`

Text Solution

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Since `vecd` makes equalw angles with the vectors `veca1 , vecb and vecc`, we have,
`d= (mu(veca + vecb + vecc))/3`
(`vecd` passes through the centroid of the triangle with position vectors, `veca , vecb and vecc`)
Again `[veca vecb vecc]vecd = [ vecd vecb vecc] + [vecd vecc vecd] vecb`
`+ [vecd veca vecb]vecc`
From (i) and (ii) , we get `[veca vecb vecc] = [vecd vecc veca] = [ vecd veca vecb] `
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Knowledge Check

  • 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 .

    A
    Both the statements are true and statement 2 is the correct explanation for statement 1.
    B
    Both statements are true but statement 2 is not the correct explanation for statement 1.
    C
    Statement 1 is true and Statement 2 is false
    D
    Statement 1 is false and Statement 2 is true.

  • Let veca, vecb and vecc be three mutually perpendicular unit vectors. If vecd is a linear combination of veca, vecb and vecc such that vecd makes equal acute angles with all three vectors veca, vecb and vecc and |vecd|=2 , then the value of |veca + vecb, vecc + vecd| is

    A
    `sqrt(3) +2`
    B
    `sqrt(3)-1`
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    `sqrt(2+sqrt(5))`
    D
    `sqrt(7 + 2sqrt(3))`

  • If veca, vecb, vecc, vecd are coplanar vectors, then (vecaxxvecb)xx(veccxxvecd)=

    A
    `1`
    B
    `veca`
    C
    `vecb`
    D
    `vec0`

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