Scalar triple product Another operation with vectors is the scalar triple product, defined to be u ∙ ( v × w ), for nonzero vectors u , v , and win ℝ 3 . 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.
Scalar triple product Another operation with vectors is the scalar triple product, defined to be u ∙ ( v × w ), for nonzero vectors u , v , and win ℝ 3 . 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.
Solution Summary: The author proves that the position vectors u, v, and w are coplanar, only when they lie on the same plane.
Scalar triple product Another operation with vectors is the scalar triple product, defined to be u ∙ (v × w), for nonzero vectors u, v, and win ℝ3.
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.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Precalculus Enhanced with Graphing Utilities (7th Edition)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.