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assig3_2011 - Express the vector[2 1 0 as a linear...

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Math 152, Spring 2011 Assignment #3 Notes: Each question is worth 5 marks. Due in class: Monday, January 24 for MWF sections; Tuesday, January 25 for TTh sections. Solutions will be posted Tuesday, January 25 in the afternoon. No late assignments will be accepted. 1. Consider the vectors a = [2 , 0 , - 1] and b = [1 , 3 , 1]. (a) (2 marks) What is the angle between a and b ? Remember that it is better to use the dot product to determine angles between two vectors. (b) (3 marks) What is the area of the parallelogram with a and b as sides? 2. Consider the vectors a = [2 , 0 , 3], b = [ - 1 , 2 , 2] and c = [2 , 0 , 0]. (a) (4 marks) Determine whether these vectors are linearly indepen- dent. ( Hint : use a matrix with row vectors a , b and c .) (b) (1 mark) Can any vector x with three components be expressed as a linear combination of a , b and c ? Justify briefly. 3. (5 marks) Consider the vectors a , b and c given in problem 2 above.
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Unformatted text preview: Express the vector [2, 1, 0] as a linear combination of a , b and c . 4. (5 marks) Let a = [-7, -1, 0] and b = [-2, 0, 1]. Find all vectors c such that the vectors a , b and c are linearly dependent (that is they all lie on the same plane). 5. Consider the three planes P 1 , P 2 and P 3 de²ned by: P 1 : x + 2 y-z = 5 P 2 : c x + y + z =-2 P 3 : d x-2 y + h z = 11 where c , d and h are constants. (a) (2 marks) Find all constants c , d and h such that these three planes intersect in a single point. (Do not fnd the intersection point.) (b) (2 marks) For which values o± d and h are the two planes P 1 and P 3 parallel? ( Hint : use the vectors normal to these planes.) (c) (1 mark) For what value o± c are the two planes P 1 and P 2 parallel?...
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