Chapter 10: Q1E (page 541)
Find the coordinates of a position in a space.
The coordinates of the described position are\({\rm{(4,0, - 3)}}\).
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Prove the property\((ca) \times b = c(a \times b) = a \times (cb)\).
Find the two unit vectors orthogonal to both \(\langle 3,2,1\rangle \) and \(\langle - 1,1,0\rangle \).
(a) Find whether the statement (Two lines parallel to a third line are parallel) is true or false in \({R^3}\).
(b) Find whether the statement (Two lines perpendicular to a third line are parallel) is true or false in \({R^3}\).
(c) Find whether the statement (Two planes parallel to a third plane are parallel) is true or false in \({R^3}\).
(d) Find whether the statement (Two planes perpendicular to a third plane are parallel) is true or false in \({R^3}\).
(e) Find whether the statement (Two lines parallel to a plane are parallel) is true or false in \({R^3}\).
(f) Find whether the statement (Two lines perpendicular to a plane are parallel) is true or false in \({R^3}\).
(g) Find whether the statement (Two planes parallel to a line are parallel) is true or false in \({R^3}\).
(h) Find whether the statement (Two planes perpendicular to a line are parallel) is true or false in \({R^3}\).
(i) Find whether the statement (Two planes either intersect or are parallel) is true or false in \({R^3}\).
(j) Find whether the statement (Two line either intersect or are parallel) is true or false in \({R^3}\).
(k) Find whether the statement (A plane and line either intersect or are parallel) is true or false in \({R^3}\).
Prove the property\((a + b) \times c = a \times c + b \times c\).
Determine the area of the parallelogram with vertices\(A( - 2,1),B(0,4),C(4,2)\) and\(D(2, - 1)\).
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