3D Dot and Cross Product
- Author:
- Dr. Jack L. Jackson II
Visualizing Dot Products and Cross Products of 2 Three-Dimensional Vectors
In the app above adjust the vectors v and w by adjusting the component sliders or input boxes, or by moving the terminal points A and B in the 3D graph view.
Check the Angle Between Vectors box to see this angle.
Check the Projection box to see w resolved into vector components parallel and orthogonal (perpendicular) to v.
Check the Dot Product box to see the definition and two different computations of the dot product. Check the Rectangle Area box to see the absolute value of the dot product illustrated as a rectangular area on the graph. The length of this rectangle is the magnitude of v, and the width is the magnitude of the projection of w in the direction of v. The sign of the dot product is positive when the angle between the vectors is acute, it is negative when this angle is obtuse, and it is zero when the angle is right.
Check the Cross Product box to see the definition and computation of the cross product. It also displays the alternate computation of the magnitude of the cross product. Check Rectangle Area to see the magnitude of the cross product illustrated as a rectangular area. The length of this rectangle is the magnitude of v, and the width is the magnitude of the projection of w in the direction orthogonal to v. Check the Parallelogram Area to see the magnitude of the cross product illustrated as the area of a parallelogram with two adjacent sides being the two original vectors. The Unit Normal checkbox shows the unit vector in the direction of the cross product. Both this vector and the cross product vector are orthogonal to both of the original vectors, i.e. they are normal (perpendicular) to any plane containing the two vectors.