# If vec A = 2i + k and B = i + 3j - 9k then express vec B = V1 - 2V2 where vec V1 is perpendicular to B and V2 is parallel to vec A.

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If vec A = 2i + k and B = i + 3j - 9k then express vec B = V1 - 2Vwhere vec V1 is perpendicular to B and V2 is parallel to vec A.

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Given $\vec A=2\hat i+\hat k$

$\vec B$ = $\hat i+3\hat j-9\hat k$

$\vec V_2$ is parallel to $\vec A$.

$\therefore$ $\vec V_2$ = $\lambda(2\hat i+\hat k)$

$\vec V_1$ is perpendicular to $\vec B$

$\vec V_1.\vec B=0$----(1)

Let $\vec V_1$ = x$\hat i$ + y$\hat j$ + z$\hat k$

Given that $\vec B=$ V1 - 2V2

⇒ ($\hat i+3\hat j-9\hat k$) = (x$\hat i$ + y$\hat j$ + z$\hat k$) - 2$\lambda$(2$\hat i$ + $\hat k$)

⇒ $\hat i+3\hat j-9\hat k$ = (x - h$\lambda$)$\hat i$ + y$\hat j$ + (z - 2$\lambda$)$\hat k$

$\therefore$ x - 4$\lambda$ = 1, y = 3

z - 2$\lambda$ = -9------(2)

From (1), $\vec V_1,\vec B=0$

$(x\hat i+y\hat j+z\hat k).(\hat i+3\hat j-9\hat k)=0$

⇒ x + 3y - 9z = 0

⇒ 1 + 4$\lambda$ + 9 - 9(-9 + 2$\lambda$) = 0 (From(2))

⇒ 10 + 4$\lambda$ + 81 + 18$\lambda$ = 0

⇒ 14$\lambda$ = 91

⇒ $\lambda$ = 91/14 = 13/2

$\therefore$ x = 1 + 4$\lambda$ = 1 + 4 x 13/2 = -9 + 13 = 4

$\therefore$ V1 = 27$\hat i$ + 3$\hat j$ + 4$\hat k$

V2 = 13/2(2$\hat i$ + $\hat k$) = 13$\hat i$ + 13/2 $\hat k$

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