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Vector Algebra

Short Answer Type

61. Find the direction cosines of the vector joining the points A (1, 2, – 3) and B (– 1, – 2. 1). directed from A to B.

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62. If A is (a₁ , a₂ , a₃),

AB with rightwards arrow on top

is vector

straight b with rightwards arrow on top space equals space straight b subscript 1 straight i with hat on top space plus space straight b subscript 2 straight j with hat on top space plus space straight b subscript 3 straight k with hat on top

, find the coordinates of B.

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63.

If $PQ with rightwards arrow on top space equals space 3 straight i with hat on top space plus space 2 straight j with hat on top space minus space straight k with hat on top$ and the coordinates of P are (1, – 1, – 2), find the coordinates of Q.

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64. Find the values of x and y so that the vectors

2 straight i with hat on top space plus space 3 straight j with hat on top

and

straight x straight i with hat on top space plus space straight y straight j with hat on top

are equal.

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65.

Let $straight a with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top$ and $straight b with rightwards arrow on top space equals space 2 straight i with hat on top space plus space straight j with hat on top.$ Is $open vertical bar straight a with rightwards arrow on top close vertical bar space equals space open vertical bar straight b with rightwards arrow on top close vertical bar space ?$ Are the vectors $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ equal?

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66. Find a unit vector in the direction of

straight a with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space 6 space straight k with hat on top.

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67. Find unit vector in the direction of vector

straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space straight k with hat on top.

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68. Find the unit vector in the direction of the sum of the vectors

straight a with rightwards arrow on top space equals space 2 straight i with hat on top space plus space 2 space straight j with hat on top space minus space 5 space straight k with hat on top space and space space straight b with rightwards arrow on top space equals space 2 straight i with hat on top space plus space straight j with hat on top space plus 3 space straight k with hat on top

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69. If $straight a with rightwards arrow on top space equals space straight i with hat on top space plus space 2 straight j with hat on top space minus space straight k with hat on top space and space straight b with rightwards arrow on top space equals space 3 straight i with hat on top space plus space straight j with hat on top space minus space 5 space straight k with hat on top comma$ find a unit vector in the direction of $straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top$ .

Here, $straight a with rightwards arrow on top space equals space straight i with hat on top space space plus space 2 straight j with hat on top space minus space straight k with hat on top comma space space straight b with rightwards arrow on top equals space 3 space straight i with hat on top space plus space straight j with hat on top space minus space 5 space straight k with hat on top$
$therefore space space space space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top space equals space left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space minus space straight k with hat on top right parenthesis space minus space left parenthesis 3 space straight i with hat on top space plus space straight j with hat on top space minus space 5 space straight k with hat on top right parenthesis space equals space minus 2 space straight i with hat on top space plus space straight j with hat on top space plus space 4 space straight k with hat on top therefore space space space space space space open vertical bar straight a with rightwards arrow on top minus straight b with rightwards arrow on top close vertical bar space equals space square root of left parenthesis negative 2 right parenthesis squared plus left parenthesis 1 right parenthesis squared plus left parenthesis 4 right parenthesis squared end root space equals space square root of 4 plus 1 plus 16 end root space equals space square root of 21 therefore space space space space space unit space vector space in space the space direction space of space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top$
$equals space fraction numerator straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top over denominator open vertical bar straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top close vertical bar end fraction space equals space fraction numerator 1 over denominator square root of 21 end fraction space left parenthesis negative 2 space straight i with hat on top space plus space straight j with hat on top space space plus space 4 space straight k with hat on top right parenthesis equals space minus fraction numerator 2 over denominator square root of 21 end fraction straight i with hat on top space plus space fraction numerator 1 over denominator square root of 21 end fraction straight j with hat on top space plus space fraction numerator 4 over denominator square root of 21 end fraction straight k with hat on top.$