Math 230 — Summer 2023 WRITTEN HOMEWORK 1
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Math 230 — Summer 2023
WRITTEN HOMEWORK 1
DUE: 5/26/2023 11:59 PM
I. Find the given vector.
1. ev where v = ⟨4, −2, −1⟩ .
2. Unit vector in the direction opposite to v = ⟨−4, 4, 2⟩ .
II. 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).
III. 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.
IV. Use the properties of the dot product to evaluate the expression, assuming that u · v = 2, ||u|| = 1, and ||v|| = 3.
1. 2u · (3u − v).
2. (u + v) · (u − v).
V. Find the decomposition v = v ∥ + v⊥ with respect to u.
1. v = ⟨4, −1, 0⟩, u = ⟨0, 1, 1⟩ .
2. v = ⟨x,y⟩, u = ⟨1, −1⟩ .
VI. Calculate the cross product assuming that v × w = ⟨2, −1, 1⟩, u × v = ⟨1, 1, 0⟩, and u × w = ⟨0, 3, 1⟩ .
1. w × (u + v).
2. (u − 2v) × (u + 2v).
VII. Calculate ||v × w|| if ||v|| = 2, v · w = 3, and the angle between v and w is 
π
VIII. 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⟩ .
2023-06-03