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Integrals

Question

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Find x such that the four points A(4, 1, 2), B(5, x, 6) , C(5, 1, -1) and D(7, 4, 0) are coplanar.

Solution

Position space vector space of space OA with rightwards arrow on top space equals space 4 straight i with hat on top plus straight j with hat on top space plus space 2 straight k with hat on top Position space vector space of space OB with rightwards arrow on top space equals space 5 straight i with hat on top plus straight x straight j with hat on top plus 6 straight k with hat on top Position space vector space of space OD with rightwards arrow on top space equals 7 straight i with hat on top plus 4 straight j with hat on top plus 0 straight k with hat on top AB with rightwards arrow on top space equals space OB with rightwards arrow on top space minus space OA with rightwards arrow on top space space equals space 5 straight i with hat on top plus straight x straight j with hat on top plus 6 straight k with hat on top minus 4 straight i with hat on top minus straight j with hat on top minus 2 straight k with hat on top equals straight i with hat on top plus left parenthesis straight x minus 1 right parenthesis straight j with hat on top plus 4 straight k with hat on top space space space space AC with rightwards arrow on top space equals space OC with rightwards arrow on top space minus space OA with rightwards arrow on top space space space space space space equals 5 straight i with hat on top space plus straight j with hat on top minus straight k with hat on top space minus 4 straight i with hat on top minus straight j with hat on top minus 2 straight k with hat on top space equals space straight i with hat on top minus 3 straight k with hat on top space space space space space space AD with rightwards arrow on top space equals space OD with rightwards arrow on top space minus space OA with rightwards arrow on top space space space space space space space space equals 7 straight i with hat on top plus 4 straight j with hat on top plus 0 straight k with hat on top minus 4 straight i with hat on top minus straight j with hat on top minus 2 straight k with hat on top equals 3 straight i with hat on top plus 3 straight j with hat on top minus 2 straight k with hat on top

The above three vectors are coplanar

rightwards double arrow AB with rightwards arrow on top. space open parentheses AC with rightwards arrow on top space cross times space AD with rightwards arrow on top close parentheses space equals space 0 rightwards double arrow space open vertical bar table row 1 cell straight x minus 1 end cell cell space space 4 end cell row 1 0 cell negative 3 end cell row 3 3 cell negative 2 end cell end table close vertical bar space equals space 0 rightwards double arrow space 1 left parenthesis 10 plus 9 right parenthesis minus left parenthesis straight x minus 1 right parenthesis space left parenthesis negative 2 plus 9 right parenthesis plus 4 left parenthesis 3 minus 0 right parenthesis space equals space 0 rightwards double arrow 9 minus 7 left parenthesis straight x minus 1 right parenthesis plus 12 space equals 0 rightwards double arrow negative 7 left parenthesis straight x minus 1 right parenthesis space equals space minus 21 rightwards double arrow straight x minus 1 space equals space 3 therefore space space straight x space equals space 4

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