answer:The cross product of vectors vec{u} and vec{v}, denoted as vec{u} times vec{v}, can be calculated using the determinant of a matrix formed by the unit vectors hat{i}, hat{j}, hat{k}, and the components of the vectors. For the given vectors, the cross product is: vec{u} times vec{v} = left| begin{array}{ccc} hat{i} & hat{j} & hat{k} 5 & 9 & 10 -2 & 6 & -9 end{array} right| Expanding the determinant, we get: vec{u} times vec{v} = (9 cdot (-9) - 10 cdot 6)hat{i} - (5 cdot (-9) - 10 cdot (-2))hat{j} + (5 cdot 6 - 9 cdot (-2))hat{k} vec{u} times vec{v} = (-81 - 60)hat{i} - (-45 + 20)hat{j} + (30 + 18)hat{k} vec{u} times vec{v} = (-141)hat{i} - (-25)hat{j} + (48)hat{k} Thus, the cross product is: vec{u} times vec{v} = left( begin{array}{c} -141 25 48 end{array} right)

answer:To find the intervals where y is increasing, we need to examine the derivative of y. begin{align} y &= (x^2 - 1)^{1/2} frac{dy}{dx} &= frac{1}{2} (x^2 - 1)^{-frac{1}{2}} cdot 2x frac{dy}{dx} &= frac{x}{ (x^2 - 1)^{frac{1}{2}}} end{align} The derivative is undefined at values {eq}x = pm 1{/eq} and the derivative is zero at {eq}x = 0{/eq}. Therefore we now have two intervals {eq}~[-infty, -1]{/eq} and {eq}~[1,infty]{/eq}. Evaluating the derivative in each interval to see if it is negative or positive begin{align} frac{dy}{dx}(-2) &= frac{-2}{ ((-2)^2 - 1)^{frac{1}{2}}}& &= -frac{2}{sqrt{3}} < 0 frac{dy}{dx}(2) &= frac{2}{ ((2)^2 - 1)^{frac{1}{2}}}& &= frac{2}{sqrt{3}} > 0 end{align} Therefore {eq}y{/eq} is increasing at interval {eq}~[1,infty]{/eq}.

answer:Since the ball is thrown vertically upward and returns to the ground at the same height, its initial and final velocities have the same magnitude but opposite directions. Therefore, the initial velocity of the ball was 20 m/s upward.

answer:The primary materials used for building the pyramids were stone, specifically limestone, which formed the main body of these monumental structures.