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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.

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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

Let OX and OY be two axes and let

straight I with hat on top space and space straight j with hat on top

be unit vectors along OX and OY respectively.

Let

OP with rightwards arrow on top space and space OQ with rightwards arrow on top

be two vectors such that

angle XOP space equals space straight A space and space angle XOQ space space equals space straight B

so that

angle POQ space equals space space straight A space minus space straight B.

Draw

PL space perpendicular space OX comma space space QM space perpendicular space OX.

Let space straight k with hat on top

be the unit vector along z-axis.
Now,

OP with rightwards arrow on top space equals space OL with rightwards arrow on top space plus space LP with rightwards arrow on top

equals space left parenthesis OP space cos space straight A right parenthesis straight i with hat on top space plus space left parenthesis OP space sin space straight A right parenthesis straight j with hat on top

and

OQ with rightwards arrow on top space equals space OM with rightwards arrow on top space plus space MQ with rightwards arrow on top

equals space left parenthesis OQ space cos space straight B right parenthesis space straight i with hat on top space plus space left parenthesis OQ space sin space straight B right parenthesis space straight j with hat on top

therefore space space space OP with rightwards arrow on top space cross times space OQ with rightwards arrow on top space equals space open square brackets left parenthesis OP space cos space straight A right parenthesis space straight i with hat on top space plus space left parenthesis OP space sin space straight A right parenthesis space straight j with hat on top close square brackets space cross times space open square brackets left parenthesis OQ space cos space straight B right parenthesis straight i with hat on top plus space left parenthesis OQ space sin space straight B right parenthesis space straight j with hat on top close square brackets

equals space OP. space OQ space open square brackets left parenthesis cos space straight A right parenthesis space straight i with hat on top space plus space sin space straight A right parenthesis space straight j with hat on top close square brackets space cross times space open square brackets left parenthesis cos space straight B right parenthesis straight i with hat on top space plus space left parenthesis sin space straight B right parenthesis straight j with hat on top close square brackets equals space OP. space OQ space open square brackets left parenthesis cos space straight A space sin space straight B right parenthesis thin space left parenthesis straight i with hat on top space cross times space straight j with hat on top right parenthesis space plus space left parenthesis sin space straight A space cos space straight B right parenthesis thin space left parenthesis straight j with hat on top space cross times straight i with hat on top space right parenthesis close square brackets space equals space OP. space OQ space left square bracket cos space straight A space sin space straight B space straight k with hat on top space minus space sin space straight A space cos space straight B space straight k with hat on top right square bracket space equals space OP. space OQ space left square bracket cos space straight A space sin space straight B space minus space sin space straight A space cos space straight B right square bracket space straight k with hat on top space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis

Also,

OP with rightwards arrow on top space cross times space OQ with rightwards arrow on top space equals space open vertical bar OP with rightwards arrow on top close vertical bar space open vertical bar OQ with rightwards arrow on top close vertical bar space sin space left parenthesis straight A minus straight B right parenthesis space left parenthesis negative straight k with hat on top right parenthesis

equals space space minus OP. space OQ space sin space left parenthesis straight A minus straight B right parenthesis space straight k with hat on top

...(2)
From (1) and (2), we get

cos space straight A space sin space straight B space minus space sin space straight A space cos space straight B space equals space minus sin space left parenthesis straight A minus straight B right parenthesis

therefore space space space space space space space space space space space space space space space space space space

sin left parenthesis straight A minus straight B right parenthesis space equals space sin space straight A space cos space straight B space minus space cos space straight A space sin space straight 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.

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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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