Cross Product Basics
What rule determines the direction of a cross-product?
Right-Hand Rule
The vectors <2,0,0> and <0,3,0> form adjacent sides of a parallelogram. Find its area.
6
What formula gives the volume of a parallelepiped?
|a . (b x c)|
What is the equation of torque?
r x F
Many amusement parks use giant swinging pendulum rides. At Six Flags, this ride swings riders over 170 feet in the air wile generating enormous torque around a pivot point.
Wonder Woman Lasso of Truth
A cross product has magnitude 24. One vector has magnitude 6 and the angle between the vectors is 30 degrees. Find the magnitude of the other vector.
8
A triangular sail is supported by vectors a and b. What formula gives its area?
1/2|a x b|
3 vectors are mutually perpendicular with lengths 2, 3, and 4. Find the volume.
24 cubic units
Why is it easier to loosen a bolt with a longer wrench?
Why are riders seated far from the center on many spinning rides?
A larger radius creates larger rotational effects and greater speed.
If 2 vectors are parallel, what is their cross product?
The zero vector
A civil engineer is designing two triangular support plates for a bridge. For Plate A, the two support beams meet at a 30 degree angle. For Plate B, the two support beams have the same lengths but meet at a 90 degree angle. Which plate has the larger area?
Plate B
A shipping container frame is modeled by:
a = <1,0,0>
b = <0,2,0>
c = <0,0,5>
Find the volume.
10 cubic units
A force is applied directly toward a pivot point. What is the torque?
Zero
A cyclist pushes straight down on a pedal when the crank arm is horizontal. Why is this position ideal?
The force is perpendicular to the crank arm, maximizing torque.
What does the magnitude of the a cross product represent?
The area of the parallelogram formed by the vectors
An engineer is designing a triangular emergency landing pad on a mountainside. The 3 corners are A(1,2,0), B(5,3,2), C(3,7,4). The area determines the amount of anti-slip coating needed. Find the area.
AB = <4,1,2> and AC = <2,5,4>
Area = 3sqrt(14)
A shipping company models a storage container using 3 vectors. 2 of the vectors are fixed. The third vector is gradually rotated until it becomes parallel to one of the other vectors. What happens to the volume of the parallelepiped?
The volume approaches zero.
A mechanic pushes with 20 N perpendicular to a wrench 0.5 m long. Find the torque magnitude.
10 Nm
If Kingda Ka pulls with a force of 50,000 N perfectly parallel to its radius vector, but El Toro pulls with a smaller force of 40,000 N perfectly perpendicular to its radius vector, which ride generates more torque?
Kingda Ka generates zero torque in that direction, meaning El Toro generates more.
Compute the cross product a x b, where a = <1,6,0> and b = <-2,5,0>.
<0,0,17>
An architect models a roof panel using two vectors. The lengths of the vectors remain unchanged, but the angle between them decreases from 80 degrees to 20 degrees. What happens to the area of the roof panel?
The area decreases.
Determine if the vectors u = <1,4,-7>, v = <2,-1,4>, and w = <0,-9,18> lie in the same plane.
The dot product results in zero. Therefore, all 3 vectors lie in the same flat plane.
A spacecraft fires a thruster, producing a very large force. However, the line of action of the force passes exactly through the spacecraft's center of mass. The force is enormous, but engineers calculate that the torque is zero. How is this possible?
Torque depends on both force and distance from the pivot (lever arm). If the force acts through the center of mass, the lever arm is zero, so torque must be zero as well.
If a boat is moving forward along the positive x-axis (v=<1,0,0>), in which direction should engineers orient the magnetic field to ensure the resulting stopping force points directly backward along the negative x-axis (-i)?
It is physically impossible.