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assig3

# assig3 - a =[1,1 and b =[1-1(a(3 marks Write[5,1 as a...

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Math 152, Spring 2010 Assignment #3 Notes: Each question is worth 5 marks. Due in class: Monday, January 25 for MWF sections; Tuesday, January 26 for TTh sections. Solutions will be posted Tuesday, January 26 in the afternoon. No late assignments will be accepted. 1. Consider the vectors a = [1, 1, 3] and b = [2,3,-2]. (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 = [1, 0, 4], b = [2,-1, 0] and c = [8,-3, 8] (a) (2 marks) Form a matrix A with row vectors a , b and c . Find the determinant of A by hand. (b) (2 marks) What MATLAB commands would you use to check your calculations above? Write out the commands by hand: it is not important to get MATLAB syntax exactly right. Note that there are descriptions of MATLAB commands in your online notes. (c) (1 mark) Based on your answer to part (a), determine whether these vectors lie on a plane. Justify briefly. 3. Consider the vectors

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Unformatted text preview: a = [1,1] and b = [1,-1]. (a) (3 marks) Write [5,1] as a linear combination of a and b , that is ﬁnd constants c 1 and c 2 such that [5 , 1] = c 1 a + c 2 b (b) (2 marks) Can any vector x with two components be expressed as a linear combination of a and b ? Justify brieﬂy. 4. Let a = [2, 2, 2] and b = [3, 0, 1]. Find all vectors c such that the vectors a , b , c are linearly dependent (that is they all lie on the same plane). 5. Consider the two lines below, given in paramtetric form L 1 : x = (0 , 1 , 2) + s (1 , , 2) L 2 : x = (4 , 2 , c ) + t (-2 , , d ) where c and d are constants. (a) (1 mark) For what value of d are the lines parallel (in the same direction)? (b) (1 mark) With the value of d above, for what value(s) of c (if any) are the two lines identical? Justify brieﬂy. (c) (3 marks) For the case c = 5 and d = 0 ﬁnd the point P on L 1 and Q on L 2 such that the distance between P and Q is as small as possible....
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