Exam 11: Analytic Geometry In Three Dimensions

The brakes on a bicycle are applied by using a downward force of p pounds on the pedal when the six-inch crank makes a angle with the horizontal.Vectors representing the position of the crank and the force are $\mathbf { V } = \frac { 1 } { 6 } \left( - \cos 70 ^ { \circ } \mathbf { j } - \sin 70 ^ { \circ } \mathbf { k } \right)$ and $\mathbf { F } = - p \mathbf { k }$ respectively.The magnitude of the torque on the crank is given by $\| \mathbf { V } \times \mathbf { F } \|$ .Using the given information,write the torque T on the crank as a function of p.

Find a set of symmetric equations for the line through the point and parallel to the specified vector or line. Point: $( - 7,0,2 )$ Parallel to: $\mathbf { v } = 4 \mathbf { i } + 5 \mathbf { j } - 3 \mathbf { k }$

Find the general form of the equation of the plane passing through the point and perpendicular to the specified vector or line. Point: $( 7,8,2 )$ Perpendicular to: $\mathbf { n } = - 4 \mathbf { i } + \mathbf { j } - 5 \mathbf { k }$

Find the dot product of u and v. =(6,6,-1) =(4,-7,-10)

The vector v and its initial point are given.Find the terminal point. v = $( - 5,8,8 )$ Initial point: (-4,-8,-8)

Find the magnitude of v. $\mathbf { v } = ( 9,10,9 )$

Both the magnitude and direction of the force on a crankshaft change as the crankshaft rotates.Vectors representing the position of the crank and the force are $\mathbf { V } = 0.16 \left( - \cos 30 ^ { \circ } \mathbf { j } - \sin 30 ^ { \circ } \mathbf { k } \right)$ and $\mathbf { F } = - 2000 \mathrm { k }$ respectively.The magnitude of the torque on the crank is given by $\| \mathbf { V } \times \mathbf { F } \|$ ,find the magnitude of the torque on the crank shaft using the position and data shown in the figure. $\mathbf { a } = 2000 , \mathbf { b } = 0.16$

Find the general form of the equation of the plane passing through the point and perpendicular to the specified line.[Be sure to reduce the coefficients in your answer to lowest terms by dividing out any common factor.] (-7,-8,-1) x=3+2t y=-3-2t z=2+4t

Use the vectors v to find v×v. $\mathbf { v } = 4 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }$

Find the magnitude of v. $\mathbf { v } = ( 1 , - 3,6 )$

Use the vectors u and v to find (-2u)× v. $\mathbf { u } = 3 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } \quad \mathbf { v } = 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$

Use the triple scalar product to find the volume of the parallelepiped having adjacent edges u,v,and w. =(1,1,2) =(2,2,2) =(2,0,2) $\mathrm { a } = ( 1,1,2 ) , \mathrm { b } = ( 2,2,2 ) , \mathrm { c } = ( 2,0,2 )$

Find u × v and show that it is orthogonal to both u and v. $\mathbf { u } = ( 6 , - 1,7 ) \quad \mathbf { v } = ( 5,5 , - 1 )$

Find the angle between the two planes in degrees.Round to a tenth of a degree. 3x-4y+1z=-6 2x+1y-3z=0

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = ( 5 ) \mathbf { i } + ( 1 ) \mathbf { j } + ( - 3 ) \mathbf { k } \quad \mathbf { v } = ( - 5 ) \mathbf { i } + ( 5 ) \mathbf { j } + ( - 2 ) \boldsymbol { k }$

Find the area of the parallelogram that has the vectors as adjacent sides. =10+6+10 =5-10+15

Find the lengths of the sides of the triangle with the indicated vertices. $( 0,0,0 ) , ( 3,3,1 ) , ( 5 , - 1,2 )$

Find the standard form of the equation of the sphere with the given characteristics. Center: (-4,1,-7);radius 9

Find the dot product of u and v. =5-10 =10-6-4

Find the standard form of the equation of the sphere with the given characteristics. Endpoints of a diameter: (-1,6,1), (7,-4,-9)