Vector triple products

What is a vector?

Geometrical elements with direction and magnitude are known as vectors. A vector is represented as a line with an arrow pointing in the direction of the vector, and the length of the line denotes the vector’s magnitude. As a result, arrows represent vectors and have two points: an initial point and a terminal point. Over 200 years, the concept of vectors has changed. Vectors represent physical quantities such as movement, velocity, and acceleration.

Types of Vectors:

Based on their magnitude, direction, and connection with other vectors, vectors are classified into distinct categories. These are some types of vectors:

1. Zero Vector –

Zero Vector is denoted by 0-> (000). The zero vector has no direction and zero magnitudes. It’s also known as vector additive identity.

2. Unit Vector-

Unit vectors, also known as the multiplicative identity of vectors, represented by a^, are vectors with a magnitude equivalent to one. A unit vector’s magnitude is one. It’s most commonly used to indicate a vector’s direction.

3. Position Vector –

In three-dimensional space, position vectors are used to identify the position and way of the motion of vectors. Position vectors’ direction and magnitude can be modified with other bodies. It’s also referred to as a location vector.

4. Equal Vector –

When the equivalent components of two or more vectors are identical, they are equal. The magnitude and direction of equal vectors are both the same. Although their initial and end sites may differ, the magnitude and direction must be identical.

5. Negative Vector- 

If two vectors have the same magnitude but opposite directions, they are said to be the negative of each other. Vector A is the negative of vector B or vice versa if vectors A and B have identical magnitude but opposite orientations.

6. Parallel Vector-

If two or more vectors have the same direction but not generally the same magnitude, they are parallel vectors. The angles of similar vectors’ directions differ by 0 degrees. Antiparallel vectors are those whose direction angles differ by 180 degrees, i.e., antiparallel vectors have entirely different directions.

Cross Products of Vectors:

A matrix represents the vector components, and a determinant of the matrix defines the outcome of the cross product of the vectors. The combination of the magnitudes of the two vectors and the angular displacement between them is another method to find the cross product of two vectors, A and B.

The cross product of one vector with the cross products of the other two vectors is called the vector triple product. The relationship is as follows:

a x (b x c) = (a.c) b – (a.b) c

Because the cross product is anticommutative, this equation (also known as triple product expansion or Lagrange’s formula) can also be expressed as:

 ( a x b) x c – c x (a x b) = -(c.b)a + (c.a)b

When we simplify the vector triple product, we get BAC – CAB identity name.

Let us take these three vectors, for example, a⃗,b⃗,c⃗

The vector a’s cross product with the cross product of the vectors b⃗ and c⃗ is their Vector triple product. With the triple product, the vectors b⃗ and c⃗ are made coplanar, meaning they are on the same plane.

Vector Triple Product Properties:

Let’s pretend there are three vectors: a, b, and c. The vector triple product of a, b, and c is the cross-product of vectors such as a x (b x c) and (a x b )x c.

It can also be written as a × (b × c) = (a. c) b − (a. b) c

The a x (b x c) vector triple product is a linear combination of the two vectors enclosed in brackets.

The r= a x (b x c) vector is parallel to a vector and stays in the b and c planes. Only when the vector is exterior to the bracket on the leftmost side is the vector r = a + λb true expression.

When ‘r’ is not discovered as described in the above theory, we shift it to the left using the cross-product properties and apply the exact expression.

Therefore, (b × c) × a

= − {a × (b × c)} 

= − {(a. c) b − (a. b) c} 

= (a. b) c − (a. c) b

As a vector quantity, the vector triple product is recognised.

Conclusion:

The triple product comprises three three-dimensional vectors, commonly Euclidean vectors, in geometry and mathematics. The scalar-valued scalar triple product and, less frequently, the vector-valued vector triple product is also referred to as “triple products.” The cross product of three vectors is dealt with in vector triple product, a branch of vector algebra. The cross product of a vector with the cross product of the other two vectors yields the value of the vector triple product. As a result, it returns a vector.