Calculus 10.4 (Cross Product)

What is the cross product of a vector with itself?

Any vector crossed with itself will result in a zero vector. This can be seen in both the definition of the cross product as well as the geometric idea that two of the same vector creates a parallelogram with 0 area and so the magnitude of the cross product vector is also 0

Find the volume of the parallelepiped of vectors, a, b, and c

Combining the definition of the cross product with the definition of the dot product the volume of a parallelepiped can be derived. Basically this is combining the geometric interpretations of the cross and dot products

Magnitude of a cross product vector (geometric interpretation)

Combining the definition of the cross product with the equation for the area of a parallelogram it is found that the magnitude of the cross product is equal to the area of the parallelogram made by the original vectors

The cross product is commutative (T/F)

False

Find a vector perpendicular to the plane that passes through the points, P(1,4,6) Q(⁻2,5,⁻1) R(1,⁻1,1)

Since the cross product of these two vectors is is perpendicular to both vectors it is also perpendicular to the plane through the points, the same is true for any scalar multiple of the cross product vector

Find the area of a triangle PQR if PQ × PR = <⁻40,⁻15,15>

Since the magnitude of the cross product of vectors PQ and PR is equal to the area of the parallelogram made by the two vectors according to the geometric definition of the cross product, the area of the triangle PQR is half the magnitude of the cross product given

The cross product is also known as what?

The "vector product" since the cross product of two vectors results in a vector

Cross product (formula)

The cross product can be determined using second order determinants and the basis vectors (i,j,k). This results in a vector that is orthogonal to both original vectors.

A bolt is tightened by applying a 40 Newton force at 75° from a 0.25 meter wrench. Find the magnitude of the torque about the center of the bolt

The magnitude of a torque vector is a direct representation of the definition of the cross product as that is precisely what it is, a cross product of position and force

Magnitude of a torque vector (equation)

The magnitude of a torque vector is a direct representation of the definition of the cross product as that is precisely what it is, a cross product of position and force

Magnitude of a cross product vector (equation)

The magnitude of the orthogonal cross product vector of two vectors is derived from the magnitudes of both original vectors multiplied by sine of the angle between them

Is torque a vector or a scalar?

Torque is a vector as it is calculated by the cross product of the position and force vectors acting on a body

The cross product of two vectors can only be defined with vectors in 3 dimensions (T/F)

True

Show that vector a × b is orthogonal to both vectors (proof)

True, using the definition of the cross product and definition of the dot product it is clear that the cross product vector is orthogonal to both original vectors

Vector a × b is orthogonal to both vectors a and b (T/F)

True, using the definition of the cross product and definition of the dot product it is clear that the cross product vector is orthogonal to both original vectors

How can it be proved that two vectors are parallel?

Two vectors are parallel if their cross product equals 0. This is because in the definition of the cross product if the angle between the vectors is 0 (as in parallel vectors) the magnitude of the resultant vector is 0 since sine of 0° = 0

Show that the vectors, a = <1,4,⁻7> b = <2,⁻1,4> c = <0,⁻9,18> are coplanar

Using the triple product / volume of the parallelepiped equation, the area of the 3D shape is found to be 0 which means that the vectors share the same plane

How can it be proved that 3 vectors are coplanar (lie in the same plane)

Using the volume of the parallelepiped equation, if the result is negative then the vectors create a 3D shape of 0 volume and so must share the same plane

j × k

i

k × i

j

i × j

k

How is a torque vector calculated?

torque is defined to be the cross product of the position and force vectors acting on a rigid body

k × j

⁻i

i × k

⁻j

j × i

⁻k