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Unformatted text preview: MATH 101 Problem Sets Note: Problems with asterisks represent supplementary informations. You may want to read their solutions if you like, but you don’t need to work on them. Set 1 1. Check if vectora bardbl vector b (a) vectora = vector i + 2 vector j vector k, vector b = 2 vector i 4 vector j + 2 vector k (b) vectora = vector i + vector j, vector b = vector j + vector k 2. Find the dot product and the angle between the two vectors vectora = vector i + vector j vector k, vector b = 2 vector i 3 vector j + 4 vector k 3. Determine whether vector PQ and vector PR are perpendicular P = ( 1 , 3 , 0) , Q = (2 , , 1) , R = ( 1 , 1 , 6). 4. Find vectora × vector b and vector c · ( vectora × vector b ) vectora = vector i + vector j + vector k, vector b = vector i vector k, vector c = vector i + vector j vector k . 5. Let vectoru and vectorv be adjacent sides of a paralelogram. Use vectors to show that the parallegram is a rectangle if the diagonals are equal in length. 6. Consider the twodimensional xyplane. Let O be the origin and A be a point lying on the xaxis. Let P be an arbitrary point of a curve. If the angle negationslash OPA ( P being the vertex) is always a right angle, what is the geometical object traced by P (the curve)? 1 Set 2 1. What is the area of the triangle which has vertices at (2 , 1 , 2) , (3 , 3 , 3) , (5 , 1 , 2)? 2. Find a vector equation, and the parametric equations for the line that contains the point ( 2 , 1 , 0), and is parallel to the vector 3 vector i 2 vector j + vector k . 3. Find an equation of the plane that contains the point ( 1 , 1 , 3) and has normal vector 2 vector i + 15 vector j 1 2 vector k . 4. Find an equation of the plane that contains the points (2 , 1 , 4), (5 , 2 , 5), and (2 , 1 , 3). 5. Show that the vector a vector i + b vector j + c vector k is a normal to the plane ax + by + cz = d where a, b, c, d are constants. 6. Use vectors to show that for any triangle the three lines drawn from each vertex to the midpoint of the opposite side all pass through the same point. 2 Set 3 1. Determine the component functions and domain of the given function (a) vector F = √ t + 1 vector i + √ 1 t vector j + vector k (b) vector F ( t ) = ( t vector i + vector j ) × ( ln ( t ) vector j + 1 t vector k ) 2. Sketch the curve traced out by the vectorvalued function. Indicate the direction in which the curve is traced out. (a) vector F ( t ) = t vector i + t vector j + t vector k (b) vector F ( t ) = cos t vector i + sin t vector j + t vector k 3. Compute the limits or explain why it does not exist (a) lim t → ( sin t t vector i + ( t + √ 2) vector j + ( e t 1) t vector k ) (b) lim t → vector F ( t ) where vector F ( t ) = t vector i + e 1 /t 2 vector j + t 2 vector k for t negationslash = 0 1 2 vector j for t = 0 4. Find the derivatives of the function (a) vector F ( t ) = t 2 cos t vector i + t 3 sin t vector j + t 4 vector k (b) vector F × vector G where vector...
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 Fall '09
 CHAN
 Multivariable Calculus, Vector Calculus, Sets, lim, Vector field, Stokes' theorem

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