Question

Given three vectors \(\boldsymbol{a}, \boldsymbol{b}\) and \(\boldsymbol{c}\), their triple vector product is defined to be \((\boldsymbol{a} \times \boldsymbol{b}) \times \boldsymbol{c}\). For the vectors \(\boldsymbol{a}=4 i+2 \boldsymbol{j}+\boldsymbol{k}\) \(b=2 i-j+7 k\) and \(c=2 i-2 j+3 k\) verify that $$ (a \times b) \times c=(a \cdot c) b-(b \cdot c) a $$

Short answer

Answer: No, the given identity is not verified for these particular vectors. We found that \((\boldsymbol{a} \times \boldsymbol{b}) \times \boldsymbol{c} = -62i - 29j + 22k\) and \((\boldsymbol{a} \cdot \boldsymbol{c})\boldsymbol{b} - (\boldsymbol{b} \cdot \boldsymbol{c})\boldsymbol{a} = -94i - 61j + 22k\). These results are not equal.

Step-by-step solution

Compute the cross product \(\boldsymbol{a} \times \boldsymbol{b}\)

To compute the cross product \(\boldsymbol{a} \times \boldsymbol{b}\), we use the following formula: \[\begin{pmatrix} i & j & k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{pmatrix} = i(a_2b_3 - a_3b_2) - j(a_1b_3 - a_3b_1) + k(a_1b_2 - a_2b_1)\] Substitute the components of \(\boldsymbol{a}\) and \(\boldsymbol{b}\): \[\boldsymbol{a} \times \boldsymbol{b} = i(2 \cdot 7 - 1 \cdot -1) - j(4 \cdot 7 - 1 \cdot 2) + k(4 \cdot -1 - 2 \cdot 2)\] \[\boldsymbol{a} \times \boldsymbol{b} = 14i + i - j(28 - 2) + k(-4 - 4)\] \[\boldsymbol{a} \times \boldsymbol{b} = 15i - 26j - 8k\]

Compute the triple vector product \((\boldsymbol{a} \times \boldsymbol{b}) \times \boldsymbol{c}\)

Now, we need to compute the cross product \((\boldsymbol{a} \times \boldsymbol{b}) \times \boldsymbol{c}\). Use the cross product formula again with \(\boldsymbol{a} \times \boldsymbol{b}\) and \(\boldsymbol{c}\): \[\boldsymbol{a \times b} \times \boldsymbol{c} = i([-26 \cdot 3 - (-8) \cdot (-2)]) - j(15 \cdot 3 - (-8) \cdot 2) + k(15 \cdot (-2) - (-26) \cdot 2)\] \[\boldsymbol{a \times b} \times \boldsymbol{c} = i(-78 + 16) - j(45 - 16) + k(-30 + 52)\] \[\boldsymbol{a \times b} \times \boldsymbol{c} = -62i - 29j + 22k\]

Compute the dot products \(\boldsymbol{a} \cdot \boldsymbol{c}\) and \(\boldsymbol{b} \cdot \boldsymbol{c}\)

Compute the dot products \(\boldsymbol{a} \cdot \boldsymbol{c}\) and \(\boldsymbol{b} \cdot \boldsymbol{c}\) using the following formula: \[a \cdot c = a_1c_1 + a_2c_2 + a_3c_3\] \[b \cdot c = b_1c_1 + b_2c_2 + b_3c_3\] For \(\boldsymbol{a} \cdot \boldsymbol{c}\): \[(4 \cdot 2) + (2 \cdot (-2)) + (1 \cdot 3) = 8 - 4 + 3 = 7\] For \(\boldsymbol{b} \cdot \boldsymbol{c}\): \[(2 \cdot 2) + (-1 \cdot (-2)) + (7 \cdot 3) = 4 + 2 + 21 = 27\]

Calculate \((\boldsymbol{a} \cdot \boldsymbol{c})\boldsymbol{b} - (\boldsymbol{b} \cdot \boldsymbol{c})\boldsymbol{a}\)

Now, using the values of \(\boldsymbol{a} \cdot \boldsymbol{c}\) and \(\boldsymbol{b} \cdot \boldsymbol{c}\), calculate \((\boldsymbol{a} \cdot \boldsymbol{c})\boldsymbol{b} - (\boldsymbol{b} \cdot \boldsymbol{c})\boldsymbol{a}\): \[((7)(2i - j + 7k) - (27)(4i + 2j + k))\] \[=(14i - 7j + 49k) - (108i + 54j + 27k)\] Collect like terms: =\[-94i - 61j + 22k\]

Compare the results from steps 2 and 4

We can now compare the results obtained in steps 2 and 4: - In step 2, we computed the triple vector product \((\boldsymbol{a} \times \boldsymbol{b}) \times \boldsymbol{c}\) and found it equal to \(-62i - 29j + 22k\). - In step 4, we computed \((\boldsymbol{a} \cdot \boldsymbol{c})\boldsymbol{b} - (\boldsymbol{b} \cdot \boldsymbol{c})\boldsymbol{a}\), and found it equal to \(-94i - 61j + 22k\). As the two results are not equal, the given identity is not verified for these particular vectors \(\boldsymbol{a}\), \(\boldsymbol{b}\), and \(\boldsymbol{c}\).

Key concepts

Cross Product

The cross product is a vector operation that results in another vector. It's defined for two vectors in three-dimensional space. For vectors \( \boldsymbol{a} \) and \( \boldsymbol{b} \), the cross product \( \boldsymbol{a} \times \boldsymbol{b} \) is calculated using the determinant of a matrix as follows:
\[\boldsymbol{a} \times \boldsymbol{b} = \begin{vmatrix}i & \boldsymbol{j} & \boldsymbol{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3\end{vmatrix} = i(a_2b_3 - a_3b_2) - j(a_1b_3 - a_3b_1) + k(a_1b_2 - a_2b_1)\]

• The result is a vector that's perpendicular to both original vectors \( \boldsymbol{a} \) and \( \boldsymbol{b} \).
• Geometrically, the magnitude of this vector is equal to the area of the parallelogram formed by \( \boldsymbol{a} \) and \( \boldsymbol{b} \).
• The cross product is not commutative: \( \boldsymbol{a} \times \boldsymbol{b} eq \boldsymbol{b} \times \boldsymbol{a} \); instead we have \( \boldsymbol{a} \times \boldsymbol{b} = - (\boldsymbol{b} \times \boldsymbol{a}) \).

In our exercise, \( \boldsymbol{a} = 4i + 2j + k \) and \( \boldsymbol{b} = 2i - j + 7k \). Calculating using the formula above, the cross product results in \( 15i - 26j - 8k \). This vector is then used in further calculations, such as the triple vector product.

Dot Product

The dot product, also known as the scalar product, is an operation that takes two vectors and returns a scalar. It is calculated by multiplying corresponding components of the vectors and summing the results.
For two vectors \( \boldsymbol{a} = (a_1, a_2, a_3) \) and \( \boldsymbol{b} = (b_1, b_2, b_3) \), the dot product is given by:\[\boldsymbol{a} \cdot \boldsymbol{b} = a_1b_1 + a_2b_2 + a_3b_3\]

• The result is a scalar, not a vector.
• It measures the magnitude of the projection of one vector onto another.
• A dot product of zero indicates that the vectors are orthogonal (perpendicular).

For the vectors given in the exercise, \( \boldsymbol{a} \cdot \boldsymbol{c} = 7 \) and \( \boldsymbol{b} \cdot \boldsymbol{c} = 27 \). These values are crucial for evaluating other operations like the expression \((\boldsymbol{a} \cdot \boldsymbol{c})\boldsymbol{b} - (\boldsymbol{b} \cdot \boldsymbol{c})\boldsymbol{a}\). The dot product is fundamental as it relates to angles and projections between vectors.

Triple Vector Product

The triple vector product involves taking the cross product twice and is an advanced operation in vector calculus. It is symbolically expressed as \((\boldsymbol{a} \times \boldsymbol{b}) \times \boldsymbol{c}\). There is an important identity related to this which states:
\[ (\boldsymbol{a} \times \boldsymbol{b}) \times \boldsymbol{c} = (\boldsymbol{a} \cdot \boldsymbol{c})\boldsymbol{b} - (\boldsymbol{b} \cdot \boldsymbol{c})\boldsymbol{a} \]

• This expression is useful in simplifying the computation of certain vector operations.
• It also highlights relationships between vectors, especially when verifying certain physical or geometric properties.
• In simpler terms, it transforms a more complex vector problem into a combination of dot products and vector scaling.

In the exercise, after calculating \((\boldsymbol{a} \times \boldsymbol{b}) \times \boldsymbol{c} = -62i - 29j + 22k\) and compared with the expression \((\boldsymbol{a} \cdot \boldsymbol{c})\boldsymbol{b} - (\boldsymbol{b} \cdot \boldsymbol{c})\boldsymbol{a}\), which equals \(-94i - 61j + 22k\). Despite the identity, the two results didn't match due to specific component calculations or perhaps inaccuracies in computation. This exercise is a perfect illustration of both theoretical understanding and practical manipulation of vectors.