The Cross Product and the Triple Product

Name: The Cross Product and the Triple Product
Author: StemPy Lab

A wrench turns a bolt best when you push at right angles to the handle. The cross product captures this: it is a vector at right angles to both of its factors, and its size is the area of the parallelogram they span. In this course you use the right-hand rule and find areas of triangles and parallelograms. Then you use the scalar triple product for volumes, and for the test of whether four points lie in one plane. It covers the last part of the Indian syllabus unit (NCERT, JEE Main) and the matching topics of A-Level Further Maths, IB Higher Level and AP.

3 lessons, about 1 h 15 min3 simulations10 problems3 practice questionsPython in every lesson

Created by StemPy Lab

Course outline (3 lessons)

Lesson 1about 25 min
The cross product and the right-hand rule
The cross product $\vec a\times\vec b$ is a vector at right angles to both $\vec a$ and $\vec b$ , with the size $\lvert\vec a\rvert\lvert\vec b\rvert\sin\theta$ . The right-hand rule gives its direction. In components it is a determinant. It changes sign when the two vectors swap places.
1 simulation1 practice question1 Python program3 problems
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Lesson 2about 25 min
Areas from the cross product
The size of $\vec a\times\vec b$ is the area of the parallelogram on $\vec a$ and $\vec b$ , so a triangle has half of it. This gives the area of a triangle from its vertices, the area of a parallelogram from its diagonals, a test for three points on a line and the identity of Lagrange.
1 simulation1 practice question1 Python program3 problems1 Python lab
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Lesson 3about 25 min
The scalar triple product, volumes and coplanarity
The scalar triple product $\vec a\cdot(\vec b\times\vec c)$ is a determinant. Its size is the volume of the parallelepiped on the three vectors, and it is zero exactly when the three vectors lie in one plane. This tests four points for coplanarity and gives the volume of a tetrahedron.
1 simulation1 practice question1 Python program4 problems1 Python lab1 question set1 note
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The Cross Product and the Triple Product: Chapter Test

20 questions · 50 min

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The Cross Product and the Triple Product

Course outline (3 lessons)

The cross product and the right-hand rule

Areas from the cross product

The scalar triple product, volumes and coplanarity

Test Your Learning

Your teacher

Discussion (0)