Solve Coplanar Vector Expression: \((\vec a\times \vec b)\cdot\vec c\)

We are asked to evaluate the expression \((2\vec a\times 3\vec b)\cdot4\vec c+(5\vec b\times 3\vec c)\cdot6\vec a\), given that the vectors \(\vec a, \vec b, \vec c\) are coplanar. This problem involves understanding vector operations, specifically the cross product and the dot product, which combine to form the scalar triple product.

What are Coplanar Vectors?

Three vectors are said to be coplanar if they lie in the same plane. Geometrically, this means that if you place the initial points of the three vectors at the same origin, their terminal points and the origin all lie on a single plane.

Understanding the Scalar Triple Product (STP)

The scalar triple product of three vectors \(\vec u, \vec v, \vec w\) is defined as \((\vec u \times \vec v) \cdot \vec w\). It is denoted by \([\vec u, \vec v, \vec w]\). The absolute value of the scalar triple product represents the volume of the parallelepiped formed by the three vectors as adjacent edges.

A key property related to coplanar vectors is that the scalar triple product of three coplanar vectors is always zero. This is because the volume of the parallelepiped formed by coplanar vectors is zero (they lie flat on a plane).

So, if \(\vec a, \vec b, \vec c\) are coplanar, then \([\vec a, \vec b, \vec c] = (\vec a \times \vec b) \cdot \vec c = 0\).

Another important property of the scalar triple product is its cyclic permutation property:

• \([\vec a, \vec b, \vec c] = [\vec b, \vec c, \vec a] = [\vec c, \vec a, \vec b]\)

However, swapping any two vectors negates the result:

• \([\vec b, \vec a, \vec c] = -[\vec a, \vec b, \vec c]\)

Step-by-Step Evaluation of the Vector Expression

The given expression is \((2\vec a\times 3\vec b)\cdot4\vec c+(5\vec b\times 3\vec c)\cdot6\vec a\). Let's evaluate each term separately.

Simplifying the First Term: \((2\vec a\times 3\vec b)\cdot4\vec c\)

First, consider the cross product \(2\vec a\times 3\vec b\). Using the property \(k(\vec u \times \vec v) = (k\vec u) \times \vec v = \vec u \times (k\vec v)\) and \((k_1\vec u) \times (k_2\vec v) = (k_1 k_2) (\vec u \times \vec v)\):

\(2\vec a\times 3\vec b = (2 \times 3)(\vec a \times \vec b) = 6(\vec a \times \vec b)\)

Now, let's take the dot product with \(4\vec c\):

\((6(\vec a \times \vec b))\cdot(4\vec c)\)

Using the property \((k\vec u)\cdot\vec v = k(\vec u \cdot \vec v)\) or \(\vec u \cdot (k\vec v) = k(\vec u \cdot \vec v)\), we can extract the scalar coefficients:

\((6(\vec a \times \vec b))\cdot(4\vec c) = (6 \times 4)(\vec a \times \vec b)\cdot\vec c = 24(\vec a \times \vec b)\cdot\vec c\)

This is \(24\) times the scalar triple product \([\vec a, \vec b, \vec c]\).

First term = \(24[\vec a, \vec b, \vec c]\).

Simplifying the Second Term: \((5\vec b\times 3\vec c)\cdot6\vec a\)

First, consider the cross product \(5\vec b\times 3\vec c\):

\(5\vec b\times 3\vec c = (5 \times 3)(\vec b \times \vec c) = 15(\vec b \times \vec c)\)

Now, let's take the dot product with \(6\vec a\):

\((15(\vec b \times \vec c))\cdot(6\vec a)\)

Extracting the scalar coefficients:

\((15(\vec b \times \vec c))\cdot(6\vec a) = (15 \times 6)(\vec b \times \vec c)\cdot\vec a = 90(\vec b \times \vec c)\cdot\vec a\)

This is \(90\) times the scalar triple product \([\vec b, \vec c, \vec a]\).

Second term = \(90[\vec b, \vec c, \vec a]\).

Using the Coplanarity Condition for STP

The expression is the sum of the two terms:

Expression = \(24[\vec a, \vec b, \vec c] + 90[\vec b, \vec c, \vec a]\)

We know from the cyclic property of the scalar triple product that \([\vec b, \vec c, \vec a] = [\vec a, \vec b, \vec c]\).

So the expression becomes:

Expression = \(24[\vec a, \vec b, \vec c] + 90[\vec a, \vec b, \vec c]\)

Expression = \((24 + 90)[\vec a, \vec b, \vec c] = 114[\vec a, \vec b, \vec c]\)

We are given that the vectors \(\vec a, \vec b, \vec c\) are coplanar. As discussed earlier, the scalar triple product of coplanar vectors is zero.

So, \([\vec a, \vec b, \vec c] = 0\).

Calculating the Final Value

Substitute the value of the scalar triple product into the expression:

Expression = \(114 \times 0 = 0\)

Thus, the value of the given expression is 0.

Key Concepts: Coplanar Vectors and Scalar Triple Product

This problem highlights the importance of understanding:

• The definition of coplanar vectors.
• The definition and properties of the scalar triple product (\((\vec a \times \vec b) \cdot \vec c\)).
• The condition for coplanarity in terms of the scalar triple product (\([\vec a, \vec b, \vec c] = 0\)).
• Properties of scalar multiplication with cross and dot products.
• Cyclic properties of the scalar triple product.

Revision Table: Vector Coplanarity

Additional Vector Algebra Concepts

Understanding vector algebra is crucial for solving problems like this. Some related concepts include:

• Vector addition and subtraction
• Scalar multiplication of vectors
• Dot product (scalar product): \(\vec a \cdot \vec b = |\vec a| |\vec b| \cos\theta\)
• Cross product (vector product): \(\vec a \times \vec b = |\vec a| |\vec b| \sin\theta \:\hat n\)
• Geometric interpretation of dot and cross products
• Properties of dot and cross products (distributive, associative for scalar multiplication)

Being comfortable with manipulating scalar constants within vector products is also key, as shown in the simplification steps of the problem.