Let a=2 i+ j-2 k, b= i+ j. · questionr

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Question 247 of 379MediumMCQ

Let

\vec a=2\hat i+\hat j-2\hat k

,

\vec b=\hat i+\hat j

. If

\vec c

satisfies

\vec a\cdot\vec c=|\vec c|

,

|\vec c-\vec a|=2\sqrt{2}

, and the angle between

\vec a\times\vec b

and

\vec c

is

\dfrac\pi6

, then

|(\vec a\times\vec b)\times\vec c|

is:

Solutions · 1

Official solution

|\vec a|=3

. Let

c=|\vec c|

:

\vec a\cdot\vec c=c

. From

|\vec c-\vec a|^2=8

:

c^2+9-2c=8 \Rightarrow (c-1)^2=0 \Rightarrow c=1

. \[\vec a\times\vec b = \begin{vmatrix}\hat i&\hat j&\hat k\\2&1&-2\\1&1&0\end{vmatrix} = (2,-2,1),\qquad |\vec a\times\vec b|=3\] \[|(\vec a\times\vec b)\times\vec c| = |\vec a\times\vec b|\,|\vec c|\sin\frac\pi6 = 3\times1\times\frac12=\frac32\]

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