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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

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

Let OACB and OAED be two parallelograms on the same base OA and between the same parallels are equal in area.
Let

OA with rightwards arrow on top space equals space straight a with rightwards arrow on top space space and space OB with rightwards arrow on top space equals space straight b with rightwards arrow on top

Since BD is parallel to OA,

therefore space space space space space space BD with rightwards arrow on top space equals space straight x straight a with rightwards arrow on top comma space where space straight x space is space some space scalar. therefore space space space space space space space OD with rightwards arrow on top space equals space OB with rightwards arrow on top space plus space BD with rightwards arrow on top space equals space straight b with rightwards arrow on top space plus space straight x space straight a with rightwards arrow on top

Now vector area of parallelogram OAED =

OA with rightwards arrow on top space cross times space OD with rightwards arrow on top

equals space straight a with rightwards arrow on top space cross times space left parenthesis straight b with rightwards arrow on top space plus space straight x space straight a with rightwards arrow on top right parenthesis space equals space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space plus space straight x space straight a with rightwards arrow on top space cross times space straight a with rightwards arrow on top space equals space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space plus space 0 with rightwards arrow on top space equals space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow 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 space space space space space space space space space space space open square brackets because space space straight a with rightwards arrow on top space cross times space straight a with rightwards arrow on top space equals space 0 with rightwards arrow on top close square brackets space equals space OA with rightwards arrow on top space cross times space OB with rightwards arrow on top space equals space vector space area space of space parallelogram space OACB.

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

1702 Views