Section 13.4 The Cross Product

Torque Torque is a measure of how much a force acting on an object causes that object to rotate –The object rotates around an axis (the pivot point) –It depends on the force in the direction of movement and the distance the force is from the pivot point We can represent the force and the distance as vectors –The force vector is in the direction of the applied force –The distance vector points away from the pivot point

Torque Torque depends on the force in the direction of movement and the distance the force is from the pivot point pivot point

Torque pivot point

Geometric Definition If are vectors then their cross product, is a vector with the following properties –Magnitude –Direction: Perpendicular to the plane created by the given vectors We can use the “right hand rule” to determine the direction of the vector Contrast this with the dot product which produces a scalar quantity

Algebraic Definition Let then This can also be done by computing a 3x3 determinant

Applications of the Cross Product Area of a parallelogram Recall area of a parallelogram is base x height How does the cross product apply? h

Applications of the Cross Product What about the area of the triangle formed by two vectors (starting at the same point)?

Applications of the Cross Product Planes in 3 space –Fact: any 3 non-collinear points in 3 space determines a plane How can we determine the unique plane through these 3 non-collinear points, P, Q, and R? –Recall: If we have a normal vector to a plane and a point in the plane, we can determine the equation of a plane We need to find a normal vector to the plane Now use point normal form for a plane to find your equation

So why do we need both types of vector multiplication? Let’s compare and contrast Both operations need two vectors The dot product produces a scalar, the cross product a vector What happens in each case if the two vectors are perpendicular? –Dot product is zero –Cross product is maximum

The dot product can be used to project one vector on to another The cross product can be used to find a vector that is perpendicular to two given vectors The dot product generalizes to any dimension The cross product only exists in 3-space Both involve the lengths of the two input vectors and the angle between them