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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.

Here $straight A space equals space left parenthesis straight a subscript 1 comma space straight a subscript 2 comma space straight a subscript 3 right parenthesis space space rightwards double arrow space space space space space space straight P. straight V. space space of space straight A space equals space straight a subscript 1 straight i with hat on top space plus space straight a subscript 2 straight j with hat on top space plus space straight a subscript 3 straight k with hat on top$
Let B be $left parenthesis straight lambda subscript 1 comma space straight lambda subscript 2 comma space straight lambda subscript 3 right parenthesis$
$therefore space space space space space straight P. straight V. space of space straight B space equals space straight lambda subscript 1 straight i with hat on top space plus space straight lambda subscript 2 straight j with hat on top space plus space straight lambda subscript 3 straight k with hat on top$
Now, $AB with rightwards arrow on top space equals space straight P. straight V. space of space straight B space minus space straight P. straight V. space of space straight A$
$rightwards double arrow space space space straight b with rightwards arrow on top space equals space left parenthesis straight lambda subscript 1 straight i with hat on top space plus space straight lambda subscript 2 straight j with hat on top space plus space straight lambda subscript 3 straight k with hat on top right parenthesis space minus space left parenthesis straight a subscript 1 straight i with hat on top space plus space straight a subscript 2 straight j with hat on top space plus space straight a subscript 3 straight k with hat on top right parenthesis rightwards double arrow space space 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 space equals space left parenthesis straight lambda subscript 1 minus straight a subscript 1 right parenthesis space straight i with hat on top space plus space left parenthesis straight lambda subscript 2 minus straight a subscript 2 right parenthesis space straight j with hat on top space plus space left parenthesis straight lambda subscript 3 minus straight a subscript 3 right parenthesis space straight k with hat on top rightwards double arrow space space space straight lambda subscript 1 minus straight a subscript 1 space equals space straight b subscript 1 comma space straight lambda subscript 2 minus straight a subscript 2 space equals space straight b subscript 2 comma space space space space straight lambda subscript 3 minus straight a subscript 3 space equals space straight b subscript 3 rightwards double arrow space space space space straight lambda subscript 1 space equals space straight a subscript 1 space plus space straight b subscript 1 comma space space straight lambda subscript 2 space equals space straight a subscript 2 space plus space straight b subscript 2 comma space space straight lambda subscript 3 space equals space straight a subscript 3 space plus space straight b subscript 3 therefore space space space straight B space is space left parenthesis straight a subscript 1 plus straight b subscript 1 comma space straight a subscript 2 plus straight b subscript 2 comma space straight a subscript 3 plus straight b subscript 3 right parenthesis.$

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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?