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- 2. If , and the vector is drawn with its tail at the point, find the coordinates of the point at the head of .arrow_forwardWe mentioned in Section 7.5 that our algebraic treatment of vectors could be attributed, in part, to the Irish mathematician William Rowan Hamilton. Hamilton considered his greatest achievement to be the discovery of qualernions, which he (incorrectly) predicted would revolutionize physics. Research the subject of quaternions. What are they? Why did quaternions fail to be as useful for physics as Hamilton predicted? How are quaternions useful, instead, for 3.D computer graphics? Write a paragraph or two about your findings.arrow_forwardThis is a linear algebra problem from section 1.3 "Homogenous Equations". All step work is greaty appreciated!arrow_forward
- 1. Determine "k" so that the vectors 3a- 4b and 5 a+ 2k b are collinear. Justify your response.arrow_forwardSuppose the vectors u, v, x, y, z are distinct vectors in Rª, and that the vectors x, y, z are linearly independent. Which of the following are TRUE? (select all that apply) 0 The vectors x+y+z, x-y+z, x + z are linearly independent. The equation ax +by+cz = 0 has only the solution a = b = c = 0. The vectors x + y, y+z, x+z are linearly independent. One of the vectors in the set {x, y, z) can be expressed as a linear combination of the other two. Every vector in R4 can be expressed as a linear combination of the vectors x, y, z. The vectors u, v, x, y, z are linearly independent.arrow_forwardThe zero vector 0 = (0, 0, 0) can be written a linear combination of the vectors v., v,, and v- because 0 = Ov, + 0v, + 0v3. This is called the trivia! solution. Can you find a nontrivial way of writing 0 as a linear combination of the three vectors? (Enter your answer in terms of v., v,, and v- If not possible, enter IMPOSSIBLE.) V1 = (1, 0, 1), V2 = (-1, 1, 2), V3 = (0, 1, 6)arrow_forward
- This is from section 1.3 "Homogenous Equations" out of the textbook titled "Linear Algebra With Applications". If you could provide step by step instructions in how to do these problems, I would be grateful. Thanks!arrow_forwardSuppose that A= 2 6 2 [-1 1 1] Describe the solution space to the equation Ax = 0. Describe the solution space to the equation Ax = b where b : Are there any vectors b for which the equation Ax = b is inconsistent? Explain your answer. Do the columns of A span R? Explain your answer.arrow_forward1.The following problems will all make reference to these three vectors: 4 7 A= 4 B = 5 C= -1 -4 -5 a. Calculate the following dot products: Ä•B, Ã•Č¸ and B•(À+2č). b. Which vector (among A, B, and C) is the longest? Which is the shortest? C. Calculate the vector product Â×(B×ċ).arrow_forward
- 1. Let (1,2) and 7: = (a) For which value of a the two vectors an are parallel? For which value of a they are orthogonal to each other? (b) For which value(s) of a the two vectors an 2. Consider the vectors (a) Is (b) Is = (3, a) be two vectors in R². Let = (1, 1, 1, 1) (1,2,3,0), = (0,1,2,3) and = (2,3,4,-3) a linear combination of a linear combination of are linearly independent? and ?? , and ?arrow_forward4. a, Determine whether vectors V and W are parallel, orthogonal, , or neither. If parallel determine whether they have the same directions or opposite directions V= -2i+3j ( 1= 2 6 v = 31-5j v=-2i + 3j W = -61 +9 (245) j w = 61 +10 j w = -6i-4 jarrow_forwardIf a, b and c are three non-coplanar vectors, then find the value of a. (bx c) b·(c×a)¸c·(b×a) + b.(cxa) c.(axb) a (bxc) a.(bxc)arrow_forward
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