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Scalar triple product Another operation with
65. Explain why the position vectors u, v, and w are coplanar if and only if |u ∙ (v × w)| = 0. (Hint: See Exercise 63).
63. Consider the parallelepiped (slanted box) determined by the position vectors u, v, and w (see figure). Show that the volume of the parallelepiped is |u ∙ (v × w) |, the absolute value of the scalar triple product.
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Chapter 13 Solutions
Calculus: Early Transcendentals (3rd Edition)
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- Take this test to review the material in Chapters 4and Chapters 5. After you are finished, check your work against the answers in the back of the book. Write w=(7,2,4) as a linear combination of the vectors v1, v2 and v3 if possible. v1=(2,1,0), v2=(1,1,0), v3=(0,0,6)arrow_forwardConsider the vector v=(1,3,0,4). Find u such that a u has the same direction as v and one-half of its length. b u has the direction opposite that of v and twice its length.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_forward
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