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s2 - Solution to Set 2 1 Two of the sides of the triangle...

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Solution to Set 2 1. Two of the sides of the triangle are represented by the line segments a = (3 , 3 , 3) (2 , 2 , 2) = (1 , 2 , 1) and b = (5 , 1 , 2) (2 , 1 , 2) = (3 , 0 , 0). The area of the triangle is given by 1 2 | a × b | . Since a × b = ( i + 2 j + k ) × 3 i = 3 j 6 k , the area is 1 2 36 + 9 = 3 2 5. 2. vector form: r = 2 i + j + t (3 i 2 j + k ) − ∞ < t < parametric form: x = 2 + 3 t y = 1 2 t z = t 3. r P 0 = ( x + 1) i + ( y 1) j + ( z 3) k N · ( r P 0 ) = 2( x + 1) + 15( y 1) 1 2 ( z 3) = 0 equation: 4 x + 30 y z 31 = 0 4. (5 , 2 , 5) (2 , 1 , 4) = (3 , 3 , 1) (2 , 1 , 3) (2 , 1 , 4) = (0 , 2 , 1) Let N = (3 i + 3 j + k ) × (2 j k ) = 5 i + 3 j + 6 k equation: [( x 5) i + ( y 2) j + ( z 5) k ] · ( 5 i + 3 j + 6 k ) = 5 x + 3 y + 6 z 11 = 0 5. We are going to show that the vector ai + bj + ck is perpendicular to any line lying on the plane (or any directed line segments on the plane). Let P = ( x 1 , y 1 , z 1 ) and Q = ( x 2 , y 2 , z 2 ) be any two distinct points on a line that lies on the plane.
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