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**Unformatted text preview: **Math 1920, Prelim I Solutions September 30, 2010, 7:30 PM to 9:00 PM 1. Consider the vectors v = i + 2 j + a k and w = i + j + k . (a) (10 pts) Find all values of the number a (if any) such that v is perpendicular to w . v · w = 1 + 2 + a = 0. Solving for a we get a =- 3. (b) (10 pts) Find all the values of the number a (if any) such that the area of the parallelogram determined by v and w is equal to √ 6. v × w = i j k 1 2 a 1 1 1 = (2- a ) i- (1- a ) j- 1 k So 6 = | v × w | 2 = (2- a ) 2 + (1- a ) 2 + 1 and 0 = 2 a ( a- 3), so a = 0 or a = 3. 2. (10 pts) Find the plane through the origin perpendicular to the plane 2 x + 2 y + z = 1 and perpendicular to the vector v = (1 , 1 ,- 4). Such a plane is x + y- 4 z = 0. The vector (2 , 2 , 1) is perpendicular to the plane 2 x +2 y + z = 1. The vector (1 , 1 ,- 4) is perpendicular to the plane x + y- 4 z = 0, and since (2 , 2 , 1) · (1 , 1 ,- 4) = 2+2- 4 = 0, the normals to these two planes are perpendicular. This is the definition of the planesthe normals to these two planes are perpendicular....

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