A line passing through the point A with position vector is parallel to the vector Find the length of the perpendicular drawn on this line from a point p with vector - Wired Faculty

Question

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A line passing through the point A with position vector $straight a with rightwards arrow on top space equals space 4 straight i with hat on top plus 2 straight j with hat on top plus 2 straight k with hat on top$ is parallel to the vector $straight b with rightwards arrow on top space equals space 2 straight i with hat on top plus 3 straight j with hat on top plus 6 straight k with hat on top.$ Find the length of the perpendicular drawn on this line from a point p with vector $stack straight r subscript 1 with rightwards arrow on top space equals straight i with hat on top plus 2 straight j with hat on top plus 3 straight k with hat on top.$

Solution

Let the equation of the line be $straight r with rightwards arrow on top space equals space straight a with rightwards arrow on top space plus space straight lambda straight b with rightwards arrow on top$
$Here comma space straight a with rightwards arrow on top space equals space 4 straight i with hat on top plus 2 straight j with hat on top plus 2 straight k with hat on top straight b with rightwards arrow on top equals space 2 straight i with hat on top plus 3 straight j with hat on top plus 6 straight k with hat on top therefore space Equation space of space the space line space is space straight r with rightwards arrow on top space equals space 4 straight i with hat on top plus 2 straight j with hat on top plus 2 straight k with hat on top plus bold lambda left parenthesis 2 straight i with hat on top plus 3 straight j with hat on top plus 6 straight k with hat on top right parenthesis Let space straight L space be space the space foot space of space the space perpendicular space and space straight P space be space the space required space point space from which space we space have space to space find space the space length space of space the space perpendicular straight P open parentheses straight alpha with rightwards arrow on top close parentheses space equals straight i with hat on top plus 2 straight j with hat on top plus 3 straight k with hat on top PL with rightwards arrow on top space equals space Position space of space vector space of space straight L space minus space Position space vector space of space straight P space space space space space space equals 4 straight i with hat on top plus 2 straight j with hat on top plus 2 straight k with hat on top plus straight lambda open parentheses 2 straight i with hat on top plus 3 straight j with hat on top plus 6 straight k with hat on top close parentheses minus open parentheses straight i with hat on top plus 2 straight j with hat on top plus 3 straight k with hat on top close parentheses space space space space space space equals 3 straight i with hat on top minus straight k with hat on top plus straight lambda open parentheses 2 straight i with hat on top plus 3 straight j with hat on top plus 6 straight k with hat on top close parentheses space.... left parenthesis straight i right parenthesis$
Now
$PL with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 0 left square bracket Since space PL with rightwards arrow on top space is space perpendicular space to space straight b with rightwards arrow on top right square bracket space space open square brackets 3 straight i with hat on top minus straight k with hat on top plus straight lambda open parentheses 2 straight i with hat on top plus 3 straight j with hat on top plus 6 straight k with hat on top close parentheses close square brackets. space open parentheses 2 straight i with hat on top plus 3 straight j with hat on top plus 6 straight k with hat on top close parentheses equals 0 space rightwards double arrow open square brackets straight i with hat on top left parenthesis 3 plus 2 straight lambda right parenthesis plus straight j with hat on top left parenthesis 3 straight lambda right parenthesis plus space straight k with hat on top left parenthesis negative 1 plus 6 straight lambda right parenthesis close square brackets. space open parentheses 2 straight i with hat on top plus 3 straight j with hat on top plus 6 straight k with hat on top close parentheses space equals space 0 space rightwards double arrow left parenthesis 3 plus 2 straight lambda right parenthesis 2 plus left parenthesis 3 straight lambda right parenthesis 3 plus left parenthesis negative 1 plus 6 straight lambda right parenthesis 6 space equals space 0 space space rightwards double arrow 6 plus 4 straight lambda plus 9 straight lambda minus 6 plus 36 straight lambda space equals space 0 space space space rightwards double arrow 49 straight lambda space equals space 0 space space space space space therefore space space straight lambda space equals space 0 space space PL with rightwards arrow on top equals 3 straight i with hat on top minus straight k with hat on top space left square bracket from space left parenthesis ii right parenthesis right square bracket space space open vertical bar PL with rightwards arrow on top close vertical bar space equals open vertical bar square root of 3 squared plus left parenthesis negative 1 right parenthesis squared end root close vertical bar space space therefore space space open vertical bar PL with rightwards arrow on top close vertical bar space equals space square root of 10 space space space space space$
Length of the perpendicular drawn on the line from P = $square root of 10$

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