Dot Product
Find the dot product of <6, -4> and <-3,2>
-26
Find the cross product of <7,3,2> and <1, -1, 5>
<17, -33, -10>
Is the point (-4, -6, 2) on the line x = -2 + t, y = 3t, z = 4 + t?
Yes
If the quadratic surface equation is changed from x2 + y2 - z2 = 1 to x2 - y2 + z2 = 1, how is the graph affected?
The graph of the quadratic surface is now centered around the y-axis instead of the z-axis
Find an equation in rectangular coordinates for p = 1
x2 + y2 + z2 = 1
Are <2, -3, 1> and <-1, -1, -1> orthogonal or parallel?
Orthogonal
Find the unit vector orthogonal to <-3, 2, -5> and <1, -1,4>
<3/(59)0.5, 7/(59)0.5, 1/(59)0.5>
Write symmetric equations of the line that passes through (-2, 0, 3) and is parallel to <2, 4, -2>
(x+2)/2 = y/4 = (z-3)/-2
Find an equation for the surface of revolution formed by revolving the curve xy = 2 in the xy plane about the x-axis
Find an equation in rectangular coordinates for theta = pi/6
x - 30.5y = 0
Find the angle between <3, 4, 0> and <0, -2, 3>
116.3 degrees
Find the area of a triangle with vertices (0, 0, 0), (1, 0, 3), and (-3, 2, 0)
11/2
Do the following lines intersect? x = 4t+2, y = 3, z = -t+1, x = 2s+2, y = 2s+3, z = s+1
Yes
Find the two possible equations for the generating curve given that the equation of its surface of revolution is 8x2 + y2 + z2 = 5
z = (5-8x2)0.5
or
y = (5-8x2)0.5
Convert (4, 0 , 0) to spherical coordinates
(4, 0, pi/2)
Find the work done on an object pulled 10 feet across a floor with 85 pounds of force. The direction of the force is 60 degrees above the horizontal
425 ft-lbs
Find the volume of a parallelepiped with adjacent sides <3, 0, 0>, <0, 5, 1>, and <2, 0, 5>
75
Find the equation of the plane passing through (1, 2, 3), (3, 2, 1) and (-1, -2, 2)
-4x - 3y + 4z - 14 = 0
Find and identify traces for x2 + y2/4 + z2 = 1
xy trace: x2 + y2/4 = 1
-ellipse
xz trace: x2 + z2 = 1
-circle
yz trace: y2/4 + z2 = 1
-ellipse
Convert (6, pi/2, -6) from cylindrical to spherical coordinates
(6*20.5, pi/2, 3pi/4)
A 48,000 pound truck is parked on a 10 degree slope. The only force to overcome is gravity. Find the force perpendicular to the hill.
~47,270.8 lb
A brake is applied on a bike with a downwards force of 20lbs on the pedal when the crank makes an angle 40 degrees to the horizontal. If the crank is 6 inches, find the torque.
~7.66
Find the point of intersection of the plane 3x-y+4z=7 and the line through (5,4,-3) perpendicular to this plane
(77/13, 48/13, -23/13)
Sketch 16x2 - y2 + 16z2 = 4
*Hyperboloid of one sheet centered around the y axis*
Sketch the surface that follows 0<theta<pi/2, 0<r<2, 0<z<4 (assume all < signs mean less than or equal to)
*1/4 cylinder with a heigh from 0 to 4 and length from 0 to 2.