Vector Triple Product

Vector Triple Product is a concept in vector algebra that involves taking the cross product of three vectors. To find its value, you calculate the cross product of one vector with the cross product of the other two vectors. The result is a new vector. When we simplify this process, we get the BAC-CAB identity.

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To make sense of the vector triple product, let's break it down:

1. Cross Product:

The cross product of two vectors, a × b, yields a vector perpendicular to both a and b with its magnitude representing the area of the parallelogram formed by the two vectors. This concept lays the foundation for understanding the triple product.

2. Double Cross:

In the triple product, a × (b × c) involves performing the cross product twice. Imagine taking the first cross product, b × c, which generates a vector lying in the plane defined by b and c. Then, we perform the second cross product with a, effectively "projecting" this intermediate vector onto a plane perpendicular to a. This final result, the vector triple product, lies in the plane containing a and the original cross product b × c.

Important Formulas:

This cross-product generates a vector quantity as a result. After the simplification of the vector triple product, BAC – CAB identity name can be obtained from the result.

BAC - CAB Identity:

Let, \[\vec{a}, \vec{b}, \vec{c}\] are the three vectors. Their Vector triple product can be defined as the cross product of vector  a with the cross product of the vectors \[\vec{b}  and  \vec{c}\]

Mathematically \[\vec{a}\times (\vec{b}\times \vec{c})\]

In this vector triple product \[\vec{a}\times (\vec{b}\times \vec{c})\]

The vectors  \[\vec{b}  and  \vec{c}\] are being coplanar with the triple product.

The triple product is also perpendicular to  \[\vec{a}\]

The mathematical form is: a × (b × c) = (a.c)b - (a.b)c

where a.b and a.c are the dot products of a with b and c, respectively. This provides a powerful alternative to directly calculating the triple product.

Do you Know About the Scalar Triple Product?

The definition for the scalar triple product can be explained as it is the dot product of one of the vectors with the cross product of the other two vectors. This is also termed as the box product or mixed product.

We can explain the scalar triple product geometrically

a. (b × c)

It is the volume of the parallelepiped distinct by the three vectors shown. 

Here, the parentheses may be omitted without causing uncertainty, since the dot product cannot be estimated first. 

If it were, it would leave the cross product of a scalar and a vector, which is not defined.

Vector Triple Product Proof

We need to prove that the vector triple product is the right result generated from the cross product of

\[\vec{a}, \vec{b}  and  \vec{c}\]

\[\vec{a}\times (\vec{b}\times \vec{c})\]

This product can be written as the linear combination of vectors \[\vec{a} and \vec{b}\]

The product can be written as \[(\vec{a}\times \vec{b})\times \vec{c}=x \vec{a} + y \vec{b}\]

\[\vec{c}.(\vec{a}\times \vec{b})\times \vec{c}=\vec{c}. (x\vec{a}+y\vec{b})\]

\[x.(\vec{c}.\vec{a}) + y.(\vec{c}.\vec{b})\]

\[x.(\vec{a}.\vec{c}) + y.(\vec{b}.\vec{c}=0\]

\[\frac{x}{\vec{b}.\vec{c}}=-\frac{y}{\vec{a}.\vec{c}}=\lambda \]

So, \[x = \lambda (\vec{b}.\vec{c})\] and \[y = \lambda (\vec{a}.\vec{c})\]

Now, we have  \[(\vec{a}\times \vec{b}) \times \vec{c} = x \vec{a} + y \vec{b}\]......(1)

Let’s just substitute the value of x and y in the above equation (1)

\[(\vec{a} \times \vec{b}) \times \vec{c} = (\lambda \vec{b}. \vec{c}) \vec{a} + (-\lambda \vec{a}. \vec{c}) \vec{b} = (\lambda \vec{b}. \vec{c}) \vec{a}-(\lambda \vec{a}. \vec{c}) \vec{b}\]

The product is for each value of \[\vec{a},\vec{b} and \vec{c}\]

The reason is each of them has an identity.

Put \[\vec{a} = i, \vec{b} = j, \vec{c} = i\]

\[(i \times j) \times = (\lambda j . i) i - (\lambda i . i) j\]

\[j = - \lambda j\]

\[\lambda = -1\]

Therefore, \[(\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a} . \vec{c}) \vec{b} - (\vec{b} . \vec{c}) \vec{a}\]

Vector Triple Product Example 

Let’s solve some vector triple product example problems.

Example – 1

There are three vectors known as \[\vec{a}, \vec{b} and \vec{c}\].

The magnitudes of these vectors are \[\left | \vec{a} \right |= 1, \left | \vec{b} \right |=2, \left | \vec{c} \right |=1\]

The equation is \[\vec{a} \times (\vec{a} \times \vec{b}) + \vec{c} = 0\].

Then, calculate the angle between \[\vec{a} and \vec{b}\].

Ans: Consider the angle between \[\vec{a} and \vec{b}\] = A

We know the relation that

\[\vec{a}.\vec{b} = \left | \vec{a} \right | \left | \vec{b} \right |cos A\]

\[= 1*2*cos A\]

\[= 2\]

Also, \[\vec{a} \times (\vec{a} \times \vec{b}) + \vec{c} = 0\]

\[(\vec{a} . \vec{b}) \vec{a} - (\vec{a} . \vec{a}) \vec{b} + \vec{c} = 0\]

2cosA \[\vec{a} - \vec{b} + \vec{c} = 0\]

2cosA \[\vec{a} - \vec{b} = - \vec{c}\]

Lets just square both the sides

2cosA \[[\vec{a} - \vec{b}]^{2} = [-\vec{c}]^{2}\]

\[4cos^{2}A\left | \vec{a} \right |^{2} - 4cosA \vec{a} . \vec{b} + \left | \vec{b} \right |^{2} = \left | \vec{c} \right |^{2}\]

\[4cos^{2}A - 4cosA.2cosA + 4 = 1\]

\[4(1-A)=1\]

\[1=4sin^{2}A\]

Lets just apply square root on both sides, we get;

\[1= 2sin A\]

\[Sin A\frac{1}{2}\]

\[A = Sin^{-1}\frac{1}{2} = \frac{\pi}{6}\]

Example – 2

Let’s assume that the vectors  \[\vec{a},\vec{b}, and \vec{c}\]

are coplanar. The question is to demonstrate that  \[\vec{a} \times \vec{b}, \vec{a} \times \vec{c}, \vec{b} \times \vec{c}\]

 are also coplanar.

Ans: As \[\vec{a}, \vec{b}  and \vec{c}\]

are coplanar we can write them as 

\[[\vec{a} \times \vec{b} \times \vec{c}]\]

\[[\vec{a} \times \vec{b} \times \vec{c}]=0\]

By squaring both the sides, we get,

\[[\vec{a} \times \vec{b} \times \vec{c}]^{2}=0\]

\[[(\vec{a} \times \vec{b})(\vec{b} \times \vec{c})(\vec{c} \times \vec{a})] = 0\]

After this result, it is proven that \[\vec{a}\times \vec{b}, \vec{a} \times \vec{c}, \vec{b} \times \vec{c}\] are also coplanar.

Example – 3

When both \[\vec{a} and \vec{c}\]

are collinear, then prove that \[(\vec{a} \times \vec{b}) \times \vec{c} = \vec{a} \times (\vec{b} \times \vec{c})\].

Ans: Now, we need to assume that

\[(\vec{a} \times \vec{b}) \times \vec{c} = \vec{a} \times (\vec{b} \times \vec{c})\]

\[\Rightarrow -\vec{c} \times (\vec{a} \times \vec{b}) = \vec{a} \times  (\vec{b} \times \vec{c})\]

\[\Rightarrow (-\vec{c}.\vec{b}) \vec{a} - (-\vec{c}.\vec{a}) \vec{b} = (\vec{a} - \vec{c}) \vec{b} - (\vec{a} . \vec{b}) \vec{c}\]

\[\Rightarrow (\vec{b} . \vec{c}) \vec{a} = (\vec{a} . \vec{b}) \vec{c}\]

\[\Rightarrow \vec{a} = (\frac{ \vec{a}.\vec{b}}{\vec{b}.\vec{c}})  \vec{c}\]

\[\Rightarrow \vec{a} = \lambda \vec{c}\]

Here, \[\lambda \epsilon R\]

Vector Triple Product Formula 

The vector triple product formula can be written as:

\[\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a}.\vec{c}) \vec{b} - (\vec{a}.\vec{b}) \vec{c} (\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a}.\vec{c}) \vec{b} -  (\vec{b}.\vec{c})  \vec{a}\]

Here, \[\vec{a} \times (\vec{b} \times \vec{c}) \neq (\vec{a} \times \vec{b}) \times \vec{c}\]

Vector Triple Product Properties

Let’s assume that there are three vectors such as a, b, c.  The cross-product of the vectors such as a × (b × c) and (a × b) × c is known as the vector triple product of a, b, c.

Therefore, it can be written as, a × (b × c) = (a. c) b − (a. b) c

The vector triple product a × (b × c) is a linear combination of those two vectors which are within brackets.

The ‘r’ vector r=a×(b×c) is perpendicular to a vector and remains in the b and c plane.

The expression for the vector r = a1 + λb is factual only when the vector lies external to the bracket is on the leftmost side. 

When ‘r’ is not found as mentioned in the above theory, we first change it to the left via the properties of the cross-product and then put on the exact expression.

Therefore, (b × c) × a

= − {a × (b × c)} 

= − {(a. c) b − (a. b) c} 

= (a. b) c − (a. c) b

Vector triple product is recognized as a vector quantity.

a × (b × c) ≠ (a × b) × c

Example 5: 

Let a x b=c, b x c=a, and a, b, c be the moduli of the vectors a, b, c, then find a and b.

Solution: 

a = b × c and a × b = c

∴ a is perpendicular to b and c, and c is perpendicular to a and b.

a, b, and c are perpendicular to each other

Now, a = b × c = b × (a × b) = (b . b) a − (b . a) b or 

a =b² a − (b.a) b= b² a, {because a⊥b}

⇒1= b,

Therefore, \[c = a × b = ab sin90^{0}ń\]

Taking the moduli of both sides, c = ab, but b = 1 ⇒ c = a.

Example 6: Given these simultaneous equations for two vectors x and y.

x + y = a …..(i)

x × y = b …..(ii)

x . a = 1 …..(iii)

Find the values of x and y.

Solution: 

By multiplying (i) scalarly by a, we get

a . x + a . y = a²

∴ a . y = a² − 1 ..(iv),

{By (iii)} Again a × (x × y) = a × b or (a . y) x − (a . x) y = a × b

(a² − 1) x − y = a × b ..(v),

Adding and subtracting (i) and (v),

we get x = [a + (a × b)] / [a²] and y = a − x

Applications of Vector Triple Product

The vector triple product isn't just a mathematical curiosity; it finds practical applications in various fields:

• Classical Mechanics: It helps calculate the torque acting on a rigid body and analyse the motion of charged particles in magnetic fields.

• Electromagnetism: It comes in handy when dealing with electromagnetic fields and their interactions with matter.

• Crystallography: It plays a crucial role in understanding the arrangement of atoms in crystals and predicting their properties.

Conclusion

The Vector Triple Product is a concept in mathematics and physics that involves the cross product of three vectors. It is a useful tool in understanding the relationships between vectors in three-dimensional space. In simple terms, it helps us find a new vector by combining the cross product of two vectors with a third vector. This concept is essential in various fields, including engineering and physics, where understanding the direction and orientation of vectors is crucial. While it may seem complex at first, mastering the Vector Triple Product is valuable for solving problems and gaining insights into the geometry of vector spaces.