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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

74 Views

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

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

open parentheses straight i with hat on top plus space 3 space straight j with hat on top plus space 9 space straight k with hat on top close parentheses space space straight x 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 0 open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row 1 3 9 row 3 cell negative straight lambda end cell straight mu end table close vertical bar space equals 0 straight i with hat on top left parenthesis 3 straight mu space plus space 9 straight lambda right parenthesis minus straight j with hat on top left parenthesis straight mu minus 27 right parenthesis space plus space straight k with hat on top left parenthesis negative straight lambda minus 9 right parenthesis space equals 0 3 straight mu space plus space 9 straight lambda space equals 0 space space..... space left parenthesis straight i right parenthesis 27 minus straight mu space equals 0 space space space....... left parenthesis ii right parenthesis minus straight lambda minus 9 space equals 0 space...... space left parenthesis iii right parenthesis By space equation space left parenthesis ii right parenthesis space and space left parenthesis iii right parenthesis space straight mu space equals space 27 space and space straight lambda space equals space minus 9 straight lambda comma space straight mu space value space satisfy space the space equation space left parenthesis straight i right parenthesis so space space straight mu space equals space 27 comma space straight lambda space equals negative 9

928 Views

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

1702 Views