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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.

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

Consider $straight i with hat on top. space left parenthesis straight j with hat on top space cross times space straight k with hat on top right parenthesis space plus space straight j with hat on top. space left parenthesis straight i with hat on top space cross times space straight k with hat on top right parenthesis space plus space straight k with hat on top space. space left parenthesis straight i with hat on top space cross times space straight j with hat on top right parenthesis$
$equals space straight i with hat on top. space straight i with hat on top space plus space straight j with hat on top. space left parenthesis negative straight j with hat on top right parenthesis space plus space straight k with hat on top. space straight k with hat on top space equals space straight i with hat on top. space straight i with hat on top space minus space straight j with hat on top space. space straight j with hat on top space plus space straight k with hat on top. space straight k with hat on top$
$equals 1 minus 1 plus 1 space equals 1$

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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

0
$straight pi over 4$
$straight pi over 2$
$straight pi over 2$

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