Transcription
So, after learning about the dot product and cross product of two vectors, we will now be learning the triple products. That means involvement of three vectors.
You can have two different types of triple products. As a result of the product, one type becomes a scalar that we call the scalar triple product. It is given as three vectors A, B, and C: it is A dot B cross C. This B cross C is a vector, and when you make the dot product of that with A, that makes the entire product a scalar one.
If we try now to draw these vectors, we can have, say, this is vector B, vector C is like this, and this is vector A, not in the plane of B and C. So, the vector B and vector C gives us this kind of parallelogram. Now, if we take vector A into account, we will have drawing a line parallel to B here, drawing a line parallel to C here, making a similar parallelepiped here.
Sorry, making a similar parallelogram above this, we will get a parallelepiped. The drawing is not that good, but this is the idea. If you look at the scalar triple product, this represents the volume of this parallelepiped. This scalar triple product is the volume of this 3D structure here.
The scalar triple product obeys the rule. That means, A dot B cross C equals B dot C cross A equals C dot A cross B in cyclic order. If we break this cyclic order, then we get a minus sign.
How do we write this product in component form? In component form, it can be written as A dot B cross C equals the determinant of Ax Ay Az, Bx By Bz, Cx Cy Cz.
Now, let us come to the vector triple product. The vector triple product is represented by A cross (B cross C). That equals B times (A dot C) minus C times (A dot B).
We also note that if we have A cross (B cross C), that is minus C cross (A cross B). That becomes minus A (B dot C) plus B (A dot C). This quantity is completely different from A cross (B cross C) that we have started with.
Now, let us consider some vectors that we are habituated with that represent some interesting physical quantities, like position. In a Cartesian coordinate system, the position of a particle, let us say, is here. That can be represented by a position vector like this.
This is usually represented as r, and it has three components: x, y, and z. Those are exactly the coordinates of this particle. So, this is the x component, this is the y component, and this is the z component of the position vector: this part x, y, z.
We can also consider the displacement vector. If from the origin O we have a displacement along this direction of amount r and along this direction of a vector r prime, then we know that the resultant displacement is given as this vector that is r plus r prime.
So, the position vector of a particle in a coordinate system can be written as x x cap plus y y cap plus z z cap in a Cartesian coordinate system. The magnitude of this position vector r is given as the square root of x squared plus y squared plus z squared.
The unit vector along this position vector is given as r vector over the magnitude of this. That means, x x cap plus y y cap plus z z cap over the square root of x squared plus y squared plus z squared.
Now, if we consider an infinitesimally small displacement vector from the coordinate x, y, z to the coordinate x plus dx, y plus dy, z plus dz, then this small displacement can be represented by an infinitesimal displacement element dl that is dx x cap plus dy y cap plus dz z cap.
The separation between two points, that is the separation vector, can be given as the difference between position vectors r minus r prime. The magnitude of this separation vector is the absolute value of r minus r prime vector.
A unit vector in the direction from r prime to r can be given as curly r cap, that is curly r vector over the magnitude of curly r, that is r minus r prime vector over the absolute value of r minus r prime.
In a Cartesian coordinate system, this curly r vector, that is the separation vector, can be represented as x minus x prime x cap plus y minus y prime y cap plus z minus z prime z cap.
So, the magnitude of the separation vector is given as the square root of x minus x prime squared plus y minus y prime squared plus z minus z prime squared.
Similarly, by whatever we have learned earlier, the unit vector along this direction, that is the unit vector along the direction of the separation, can be given as x minus x prime x cap plus y minus y prime y cap plus z minus z prime z cap over the square root of x minus x prime squared plus y minus y prime squared plus z minus z prime squared.