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Vector Algebra

Short Answer Type

51. Find the sum of the vectors

straight a with rightwards arrow on top space equals space straight i with hat on top space minus space 2 straight j with hat on top space plus straight k with hat on top comma space space straight b with rightwards arrow on top space equals space minus 2 straight i with hat on top space plus space 4 straight j with hat on top space plus space 5 straight k with hat on top comma space straight c with rightwards arrow on top space equals space straight i with hat on top space minus space 6 straight j with hat on top space minus space 7 straight k with hat on top

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52. Compute the magnitude of the following vectors:

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 space semicolon space space space straight b with rightwards arrow on top space equals space 2 straight i with hat on top space minus space 7 straight j with hat on top space minus space 3 straight k with hat on top semicolon space space straight c with rightwards arrow on top space equals space fraction numerator 1 over denominator square root of 3 end fraction straight i with hat on top space plus space fraction numerator 1 over denominator square root of 3 end fraction straight j with hat on top space minus space fraction numerator 1 over denominator square root of 3 end fraction straight k with hat on top

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53. Write two different vectors having same magnitude.

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54. Write two different vector having same direction.

Let $straight a with rightwards arrow on top space equals space straight i with hat on top plus straight j with hat on top plus straight k with hat on top comma space space space space straight b with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 3 space straight k with hat on top$
Direction ratios of $straight a with rightwards arrow on top$ are 1, 1, 1 and direction ratios of $straight b with rightwards arrow on top$ are 3, 3, 3.
∴ direction cosines of $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ are
$fraction numerator 1 over denominator square root of 1 plus 1 plus 1 end root end fraction comma space fraction numerator 1 over denominator square root of 1 plus 1 plus 1 end root end fraction comma space fraction numerator 1 over denominator square root of 1 plus 1 plus 1 end root end fraction$

and $fraction numerator 3 over denominator square root of 9 plus 9 plus 9 end root end fraction comma space fraction numerator 3 over denominator square root of 9 plus 9 plus 9 end root end fraction comma space fraction numerator 3 over denominator square root of 9 plus 9 plus 9 end root end fraction space respectively.$

i.e. $fraction numerator 1 over denominator square root of 3 end fraction comma space fraction numerator 1 over denominator square root of 3 end fraction comma space fraction numerator 1 over denominator square root of 3 end fraction space and space fraction numerator 1 over denominator square root of 3 end fraction comma space fraction numerator 1 over denominator square root of 3 end fraction comma space fraction numerator 1 over denominator square root of 3 end fraction$
$therefore space space space straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ are different vectors but have the same direction.