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Under what conditions is yb+6c perpendicular to a? What does this say about the coefficients of b and c in the linear combination a x (b × c) =_b+ _c?

Under what conditions is yb+6c perpendicular to a? What does this say about the coefficients of b and c in the linear combination a x (b × c) =_b+ _c?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Problem 6. (The cross product project). The cross product for vectors in V is not associative, so some alternative way of handling expressions like a x (b x c) is sorely needed. We're going to prove the most common one here, in a few simple steps. Please review the material on the cross product in the notes ("Vectors in space", "Rotations in space", particularly the ezercises). If you use facts about the cross product from elsewhere I will count the whole problem as 0. 1. Explain why a x (b × c) must be a linear combination of b and c. 2. Under what conditions is yb+ ốc perpendicular to a? What does this say about the coefficients of b and e in the linear combination a x (b × c) = _b+ _c? 3. Explain why if u and v are perpendicular, u x v = uv. 4. Show how to find a vector ẽ = c – Xb that's perpendicular to b and is a linear combination of b and c. For this vector, show that b x c = b x ẽ = bẽ. 5. Conclude from this that if b x e = 0 then b is a scalar multiple of c or vice versa (b and c are linearly dependent). 6. Now, the mapping a → a x (b x c) is linear, so is determined by the image of three mutually perpendicular, nonzero vectors. Explain why b,ẽ and be are three mutually perpendicular nonzero vectors (assuming b x c ± 0) and then evaluate the mapping at these inputs. Why is it enough to compute b x (b x e) Can you use the quaternion product here? Why does that help? 7. Parts 1 and 2 should have narrowed down the coefficients to scalar multiples of a •c and a• b. Explain. Part 5 lets you find the scalars. What is the final formula? 8. The (*) is to note that we left out some "edge cases". It could be that b = 0. Is the identity you found true in that case? it could also happen that ẽ = 0. What does that say about bx c? is the identity true in that case?
Transcribed Image Text:Problem 6. (The cross product project). The cross product for vectors in V is not associative, so some alternative way of handling expressions like a x (b x c) is sorely needed. We're going to prove the most common one here, in a few simple steps. Please review the material on the cross product in the notes ("Vectors in space", "Rotations in space", particularly the ezercises). If you use facts about the cross product from elsewhere I will count the whole problem as 0. 1. Explain why a x (b × c) must be a linear combination of b and c. 2. Under what conditions is yb+ ốc perpendicular to a? What does this say about the coefficients of b and e in the linear combination a x (b × c) = _b+ _c? 3. Explain why if u and v are perpendicular, u x v = uv. 4. Show how to find a vector ẽ = c – Xb that's perpendicular to b and is a linear combination of b and c. For this vector, show that b x c = b x ẽ = bẽ. 5. Conclude from this that if b x e = 0 then b is a scalar multiple of c or vice versa (b and c are linearly dependent). 6. Now, the mapping a → a x (b x c) is linear, so is determined by the image of three mutually perpendicular, nonzero vectors. Explain why b,ẽ and be are three mutually perpendicular nonzero vectors (assuming b x c ± 0) and then evaluate the mapping at these inputs. Why is it enough to compute b x (b x e) Can you use the quaternion product here? Why does that help? 7. Parts 1 and 2 should have narrowed down the coefficients to scalar multiples of a •c and a• b. Explain. Part 5 lets you find the scalars. What is the final formula? 8. The (*) is to note that we left out some "edge cases". It could be that b = 0. Is the identity you found true in that case? it could also happen that ẽ = 0. What does that say about bx c? is the identity true in that case?
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