• Math Archive: Questions from 2023-08-23

    Find the given vector. 1. ev where v = <4, -2, -1>. 2. Unit vector in the direction opposite to v = <-4, 4, 2>.
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    Find a vector parametrization and parametric equations for the line with the given description. 1. Passes through P = (4, 0, 8), direction vector v = 7i + 4k. 2. Passes through (1, 1, 1) and (3, -5, 2). 3. Perpendicular to the xy plane, passes through the origin. 4. Parallel to the line through (1, 1, 0) and (0, -1, -2), passes through (0, 0, 4).
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    Determine whether the lines r1(t) = <0, 1, 1> + t<1, 1, 2> and r2(t) = <2, 0, 3> + t<1, 4, 4> intersect. If so, find the point of intersection.
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    Use the properties of the dot product to evaluate the expression, assuming that u

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    Find the decomposition v = v|| + v with respect to u. 1. v = <4, -1, 0>, u = <0, 1, 1>. 2. v = , u = <1, -1>.
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    Calculate the cross product assuming that v

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    Calculate ∥v×w∥ if ∥v∥ = 2 , v⋅w = 3 , and the angle between v and w is π/6.

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    Find the equation of the plane with the given description. 1. Passing through P = (2, -1, 4), Q = (1, 1, 1), R = (3, 1, -2). 2. Passes through P = (4, 1, 9) and is parallel to x + y + z = 3. 3. Contains the lines r1(t) = <2 + t, 1 + 2t, 3t> and r2(t) = <2 + 3t, 1 + t, 8t>.
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