IKEA - Name: (2 points) 175 section 2, Test 1, 3/8 You have...

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Name: (2 points) 175 section 2, Test 1, 3/8 You have the full class period to complete the test. If anything seems unclear, please ask. 1. Find an equation for the surface consisting of all points that are equidistant from the point (1 , 0 , 0) and the yz -plane. Simplify your answer, so that x,y,z each appear in only one term. (8 points) 2. Two unit vectors in three dimensional space ~v and ~w are at an angle of 2 π/ 3. What is ~v · ~w ? What is | ~v × ~w | ? What is ~v · ( ~v × ~w )? (5 points) 1
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3. For each of the following expressions, say whether it makes sense: never; only if all vectors are three dimensional; only if all vectors are two dimen- sional; both if the vectors are all three dimensional, and if they are all two dimensional. (2 points each) (a) ( ~v · ~v )(2 + | ~v | ). (b) ( ~v × ~w ) × (2 ~w ). (c) 3( ~v × ~v ) ~v . (d) ± ~v × ~w 1 + | ~v | ² · ~w . (e) ( ~v · ~w ) | ~v | 2 ( | ~w | ~v ). 4. Say ~u , ~v and ~w are vectors in three dimensional space such that | ~u | = 1 , | ~v | = 2 , | ~w | = 3 and ~u · ~v = ~u · ~w = ~v · ~w = 0 . Write ~u × ~v in terms of ~w . Justify your answer. ( Hint: there are two possibilities - write both ). (5 points) 2
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5. Find the volume of the parallelepiped formed by the vectors ~a = h 1 , 3 , 2 i , ~ b = h 1 , 0 , - 1 i and ~ c = h- 2 , 1 , - 2 i . (7 points) 6. Find a vector equation for the line of intersection of the plane given by y = z , and the plane given by x + 2 y - z = 4. (8 points) 3
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7. Are the lines ~ r 1 ( t ) = h 2 ,t, 3 - t i and ~ r 2 ( t ) = h 3 - t, 2 t, 3 t - 2 i parallel, skew, or do they intersect in a unique point? If the last, find the point. (7 points) 8. Find a scalar equation for the plane perpendicular to the line ~ r 1 ( t ) = h 2 ,t, 3 - t i , and that contains the line ~ r 2 ( t ) = h 3 - t, 2 t, 3 t - 2 i . (7 points) 4
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9. Find and sketch the traces of the surface
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IKEA - Name: (2 points) 175 section 2, Test 1, 3/8 You have...

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