15 find the vector equation of the line of

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15.Find the vector equation of the line of intersection of the following two planest
w=(1,1,0)+t(−1,2,1)16.Determine whether the following system of equations has a single point of intersection.uw
14x27z5=0[7][5]×214x42z=0[8][8][7]15z+5=015z=−5z=515z=13Subz=13into [5]7x21z=07x21(13)=07x7=07x=7x=1Subx=1z=13into [1]4x+y9z=04(1)+y9(13)=04+y3=0y=−1The point of intersection is(1,1,13)17. Find the shortest distance fromP(−4,2,6)to the plane2x3y+z8=0n=(A,B ,C)n=(2,3,1)Letx=0y=12x3y+z8=02(0)3(1)+z8=0z11=0z=11
Another point on the plane isQ(0,1,11)PQ=QPPQ=(4,2,6)−(0,1,11)PQ=(−4,1,5)|pro jn(PQ)|=|PQ ∙nn∙n|n¿|pro jn(PQ)|=|(4,1,5)(2,3,1)(2,3,1)(2,3,1)|∨(2,3,1)∨¿|pro jn(PQ)|=|(835)(4+9+1)|4+9+1¿|pro jn(PQ)|=|1614|14|pro jn(PQ)|=161414|pro jn(PQ)|=−4.28(two decimal places)The shortest distance betweenPand the plane is 4.28 units (to two decimalplaces)
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