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Vector Algebra

Long Answer Type

291.

If $straight a with rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top space$ and $straight b with rightwards arrow on top space equals space straight j with hat on top space minus space straight k with hat on top.$ Find a vector $straight c with rightwards arrow on top$ such that $straight a with rightwards arrow on top cross times space straight c with rightwards arrow on top space equals space straight b with rightwards arrow on top$ and $straight a with rightwards arrow on top. space straight c with rightwards arrow on top space equals space 3.$

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292.

Let $straight a with rightwards arrow on top space equals straight i with hat on top space plus space 4 space straight j with hat on top space plus space 2 space straight k with hat on top comma space space space space space straight b with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space space 7 space straight k with hat on top$ and $straight c with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space 4 space straight k with hat on top$ . Find a vector $straight d with rightwards arrow on top$ which is perpendicular to both $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top comma space space straight c with rightwards arrow on top. space straight d with rightwards arrow on top space equals space 15.$

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293. If with reference to a right handed system of mutually perpendicular unit vector unit vector

straight i with hat on top comma space straight j with hat on top comma space straight k with hat on top

express

straight beta with rightwards arrow on top

in the form

straight beta with rightwards arrow on top

in the form

straight beta with rightwards arrow on top space equals space straight beta with rightwards arrow on top subscript 1 space equals space straight beta with rightwards arrow on top subscript 1 space plus space straight beta with rightwards arrow on top subscript 2 space end subscript space where space stack straight beta subscript 1 with rightwards arrow on top space is space parallel space to space straight alpha space and space stack straight beta subscript 2 with rightwards arrow on top space is space perpendicular space to space straight alpha with rightwards arrow on top.

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Short Answer Type

294. $straight A with rightwards arrow on top comma space straight B with rightwards arrow on top comma space straight C with rightwards arrow on top$ are unit vectors, Suppose that $straight A with rightwards arrow on top. space straight B with rightwards arrow on top space equals space straight A with rightwards arrow on top. space straight C with rightwards arrow on top space equals space 0$ and the angle between B and C is $straight pi over 6.$ Prove that $straight A with rightwards arrow on top space equals plus-or-minus 2 space open parentheses straight B with rightwards arrow on top space cross times space straight C with rightwards arrow on top close parentheses.$

Since $straight A with rightwards arrow on top. space straight B with rightwards arrow on top space equals space straight A with rightwards arrow on top. space straight C with rightwards arrow on top space equals 0$
$therefore space space space space straight A with rightwards arrow on top space is space perpendicular space to space straight B with rightwards arrow on top space as space well space as space straight C with rightwards arrow on top therefore space space space straight A with rightwards arrow on top space is space parallel space to space straight B with rightwards arrow on top space cross times space straight C with rightwards arrow on top$
Let $straight A with rightwards arrow on top space equals space straight lambda open parentheses straight B with rightwards arrow on top space cross times space straight C with rightwards arrow on top close parentheses$ ...(1)
where $straight lambda$ is some scalar
$therefore space space space open vertical bar straight A with rightwards arrow on top close vertical bar space equals space open vertical bar straight lambda close vertical bar space space open vertical bar straight B with rightwards arrow on top space cross times space straight C with rightwards arrow on top close vertical bar space space space space space space space rightwards double arrow space space space open vertical bar straight A with rightwards arrow on top close vertical bar space equals space open vertical bar straight lambda close vertical bar space space open vertical bar straight B with rightwards arrow on top close vertical bar space open vertical bar straight C with rightwards arrow on top close vertical bar space space sin space straight pi over 6 space open vertical bar straight n with hat on top close vertical bar$
where $straight n with hat on top$ is perpendicular to $straight B with rightwards arrow on top space as space well space as space straight C with rightwards arrow on top.$
$rightwards double arrow space space space space space 1 space equals space open vertical bar straight lambda close vertical bar space left parenthesis 1 right parenthesis thin space left parenthesis 1 right parenthesis space space open parentheses 1 half close parentheses space left parenthesis 1 right parenthesis space space space space space space space space space space space space space space space open square brackets because space space space straight A with rightwards arrow on top comma space straight B with rightwards arrow on top comma space straight C with rightwards arrow on top space are space unit space vectors close square brackets.$
$therefore space space space space space from space left parenthesis 1 right parenthesis comma space space space we space get comma space space straight A with rightwards arrow on top space space equals plus-or-minus 2 space open parentheses straight B with rightwards arrow on top space cross times space straight C with rightwards arrow on top close parentheses$

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Long Answer Type

295.

Give space space space straight a with rightwards arrow on top space equals space 1 over 7 left parenthesis 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 6 space straight k with hat on top right parenthesis comma space space straight b with rightwards arrow on top space equals space 1 over 7 left parenthesis 3 space straight i with hat on top space minus space 6 space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis comma space space space space space space space space space space space space space space straight c with rightwards arrow on top space equals space 1 over 7 left parenthesis 6 space straight i with hat on top space plus space 2 space straight j with hat on top space minus space 3 space straight k with hat on top right parenthesis comma

straight i with hat on top comma space straight j with hat on top comma space straight k with hat on top

being a right-handed orthogonal system of unit vectors in space. Show that

straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top

in another such system.

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Short Answer Type

296. Find the value of the following:

straight i with hat on top. space open parentheses straight j with hat on top space cross times space straight k with hat on top close parentheses space plus space straight j with hat on top. space space left parenthesis straight i with hat on top space cross times space straight k with hat on top right parenthesis space plus space space straight k with hat on top. space space left parenthesis straight i with hat on top space cross times space straight j with hat on top right parenthesis

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297. Find the value of the following:

straight i with hat on top. space left parenthesis straight j with hat on top space cross times space straight k with hat on top right parenthesis space plus space straight j with hat on top space cross times space left parenthesis straight k with hat on top space cross times space straight i with hat on top right parenthesis space plus space straight k with hat on top space cross times space left parenthesis straight i with hat on top space cross times space straight j with hat on top right parenthesis space

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298. Find the value of the following:

straight i with hat on top space cross times space left parenthesis straight j with hat on top space cross times space straight k with hat on top right parenthesis space plus space straight j with hat on top space cross times space left parenthesis straight k with hat on top space cross times space straight i with hat on top right parenthesis space plus space straight k with hat on top space cross times space left parenthesis straight i with hat on top space cross times space straight j with hat on top right parenthesis

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Multiple Choice Questions

299.

Let vectors $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ be such that $open vertical bar straight a with rightwards arrow on top close vertical bar space equals 3 space space space and space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space fraction numerator square root of 2 over denominator 3 end fraction comma$ then $straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top$ is a unit vector, if the angle betweeen $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ is

$straight pi divided by 6$
$straight pi divided by 4$
$straight pi divided by 3$
$straight pi divided by 3$

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300.

If $straight theta$ is the angle between any two vectors $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top comma$ then $open vertical bar straight a with rightwards arrow on top. space straight b with rightwards arrow on top close vertical bar space equals space open vertical bar straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top close vertical bar$ when $straight theta$ is equal to