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Vector Algebra

Multiple Choice Questions

301.

The value of $straight i with hat on top. space left parenthesis straight j with hat on top space cross times space straight k with hat on top right parenthesis. space plus space straight j with hat on top. space left parenthesis straight i with hat on top space cross times space straight k with hat on top right parenthesis space plus space straight k with hat on top. space left parenthesis straight i with hat on top space cross times space straight j with hat on top right parenthesis$ is

0
-1
1
1

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302. Area of a rectangle having vertices A, B, C and D with position vectors

negative straight i with hat on top space plus space 1 half straight j with hat on top space plus space 4 space straight k with hat on top comma space space space straight i with hat on top space plus space 1 half straight j with hat on top space plus space 4 space straight k with hat on top comma space space space space straight i with hat on top space minus space 1 half straight j with hat on top space plus 4 straight k with hat on top space and space minus straight i with hat on top space minus space 1 half straight j with hat on top space plus space 4 straight k with hat on top comma

respectively is

$1 half$
1
2
2

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Long Answer Type

303. If A, B and C are the vertices of a triangle ABC, prove Sine Formula that
$fraction numerator straight a over denominator sin space straight A end fraction space equals space fraction numerator straight b over denominator sin space straight B end fraction space equals space fraction numerator straight c over denominator sin space straight C end fraction.$

$Let space BC with rightwards arrow on top space equals space straight a with rightwards arrow on top comma space space CA with rightwards arrow on top space equals space straight b with rightwards arrow on top comma space space AB with rightwards arrow on top space equals space straight c with rightwards arrow on top space so space that space straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space equals space minus straight c with rightwards arrow on top therefore space space space space straight a with rightwards arrow on top space cross times space straight a with rightwards arrow on top space plus space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space minus straight a with rightwards arrow on top cross times space straight c with rightwards arrow on top therefore space space space space 0 with rightwards arrow on top space plus space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space straight c with rightwards arrow on top space cross times space straight a with rightwards arrow on top therefore space space space space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space straight c with rightwards arrow on top space cross times space straight a with rightwards arrow on top therefore space space space space 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 space equals space open vertical bar straight c with rightwards arrow on top space cross times space straight a with rightwards arrow on top close vertical bar therefore space space straight a space straight b space sin space left parenthesis 180 degree minus straight C right parenthesis space equals space straight c space straight a space sin space left parenthesis 180 degree space minus space straight B right parenthesis therefore space space space straight a space straight b space sin space straight C space equals space straight c space straight a space sin space straight B$

Dividing both sides by a, b, c we get,
$fraction numerator sin space straight C over denominator straight c end fraction space equals space fraction numerator sin space straight B over denominator straight b end fraction comma space space space space space space space therefore space space space fraction numerator straight b over denominator sin space straight B end fraction space equals space fraction numerator straight c over denominator sin space straight C end fraction$
Similarly, $fraction numerator straight c over denominator sin space straight C end fraction space equals space fraction numerator straight a over denominator sin space straight A end fraction$ ...(1)
From (1) and (2), we have
$fraction numerator straight a over denominator sin space straight A end fraction space equals space fraction numerator straight b over denominator sin space straight B end fraction space equals space fraction numerator straight c over denominator sin space straight C end fraction.$

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304. Using vectors, prove that sin (A + B) = sin A + cos B + cos A sin B.

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305. Prove by vector method that sin (A – B) = sin A cos B – cos A sin B

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306. Prove that the parallelograms on the same base and between the same parallels are equal in area.

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307. If D, E, F are the mid points of the sides of triangle ABC, prove that

area left parenthesis increment DEF right parenthesis space equals space 1 fourth area space left parenthesis increment thin space ABC right parenthesis.

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Short Answer Type

308.

find space straight lambda space and space straight mu space if open parentheses straight i with hat on top plus space 3 space straight j with hat on top space plus space 9 space straight k with hat on top close parentheses space space straight x space space left parenthesis 3 space straight i with hat on top plus straight lambda space straight j with hat on top plus space straight mu space straight k with hat on top right parenthesis space equals space 0

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309.

If space straight a with rightwards arrow on top equals 4 space straight i with hat on top space minus straight j with hat on top plus space straight k with hat on top space and space straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space minus 2 straight j with hat on top plus space straight k with hat on top comma then space find space straight a space vector space parallel space to space the space vector space straight a with rightwards arrow on top plus straight b with rightwards arrow on top.

1199 Views

310.

If $straight a with rightwards arrow on top space equals space 2 straight i with hat on top plus straight j with hat on top plus 3 straight k with hat on top space space space and space straight b with rightwards arrow on top space equals space 3 straight i with hat on top space plus space 5 straight j with hat on top space minus space 2 straight k with hat on top comma space then space find space open vertical bar straight a with rightwards arrow on top cross times space straight b with rightwards arrow on top close vertical bar.$

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