Exam 11: Section 4: Vectors and the Geometry of Space

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Given Given   and   , find   . and Given   and   , find   . , find Given   and   , find   . .

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Sketch the graph of the cross product of the unit vector Sketch the graph of the cross product of the unit vector   . .

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Following are the vertices of the parallelogram. Find the coordinates of the fourth vertex. Following are the vertices of the parallelogram. Find the coordinates of the fourth vertex.

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Find the area of the parallelogram with the vertices Find the area of the parallelogram with the vertices   , and   . Round your answer to two decimal places. , and Find the area of the parallelogram with the vertices   , and   . Round your answer to two decimal places. . Round your answer to two decimal places.

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Given Given   and   find   . and Given   and   find   . find Given   and   find   . .

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Given Given   , find   . , find Given   , find   . .

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A child applies the brakes on a bicycle by applying a downward force of 25 pounds on the pedal when the crank makes a A child applies the brakes on a bicycle by applying a downward force of 25 pounds on the pedal when the crank makes a   angle with the horizontal (see figure). The crank is 6 inches in length. Find the torque at P. Round your answer to two decimal places.  angle with the horizontal (see figure). The crank is 6 inches in length. Find the torque at P. Round your answer to two decimal places. A child applies the brakes on a bicycle by applying a downward force of 25 pounds on the pedal when the crank makes a   angle with the horizontal (see figure). The crank is 6 inches in length. Find the torque at P. Round your answer to two decimal places.

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Find the area of a parallelogram that has the given vectors as adjacent sides. Round your answer to two decimal places. Find the area of a parallelogram that has the given vectors as adjacent sides. Round your answer to two decimal places.

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Given Given   and   find   . and Given   and   find   . find Given   and   find   . .

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Find a unit vector that is orthogonal to both Find a unit vector that is orthogonal to both   and   . and Find a unit vector that is orthogonal to both   and   . .

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Given Given   and   , find   . and Given   and   , find   . , find Given   and   , find   . .

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Find the area of the parallelogram that has the given vectors Find the area of the parallelogram that has the given vectors   and   as adjacent sides. and Find the area of the parallelogram that has the given vectors   and   as adjacent sides. as adjacent sides.

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Find the cross product of the unit vector Find the cross product of the unit vector   . .

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Find the triple scalar product Find the triple scalar product   .  . Find the triple scalar product   .

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Use the triple scalar product to find the volume of the parallelepiped having adjacent edges given by the vectors Use the triple scalar product to find the volume of the parallelepiped having adjacent edges given by the vectors

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Both the magnitude and the direction of the force on a crankshaft change as the crankshaft rotates. Find the torque on the crankshaft using the position and data shown in the figure. Round your answer to two decimal places. Both the magnitude and the direction of the force on a crankshaft change as the crankshaft rotates. Find the torque on the crankshaft using the position and data shown in the figure. Round your answer to two decimal places.

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Find the area of the triangle with the vertices Find the area of the triangle with the vertices   and   . Round your answer to two decimal places. and Find the area of the triangle with the vertices   and   . Round your answer to two decimal places. . Round your answer to two decimal places.

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Find the volume of the parallelepiped with the following vertices. Find the volume of the parallelepiped with the following vertices.

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