Question

If \( \vec{p} = 3\hat{i} - \hat{j} + 2\hat{k} \), \( \vec{q} = \hat{i} + 4\hat{j} - \hat{k} \), and \( \vec{r} = 2\hat{i} - 3\hat{j} + 5\hat{k} \), find \( \vec{p} \cdot (\vec{q} \times \vec{r}) \).

• A. \( 36 \)
• B. \( -36 \)
• C. \( 65 \)
• D. \( -65 \)

Hint:

Scalar triple product can be computed efficiently using the determinant method.

Answer 1

The Correct Option is A

To find \( \vec{p} \cdot (\vec{q} \times \vec{r}) \), we need to calculate the cross product \( \vec{q} \times \vec{r} \) first, and then take the dot product with \( \vec{p} \). Given vectors: \( \vec{p} = 3\hat{i} - \hat{j} + 2\hat{k} \), \( \vec{q} = \hat{i} + 4\hat{j} - \hat{k} \), \( \vec{r} = 2\hat{i} - 3\hat{j} + 5\hat{k} \). 

Step 1: Compute the cross product \( \vec{q} \times \vec{r} \).

	\(\hat{i}\)	\(\hat{j}\)	\(\hat{k}\)
\(\vec{q}\)	1	4	-1
\(\vec{r}\)	2	-3	5

The cross product \( \vec{q} \times \vec{r} \) is calculated as follows:

\( \vec{q} \times \vec{r} = \left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 1 & 4 & -1 \\ 2 & -3 & 5 \end{array}\right| \)

= \( \hat{i}(4 \cdot 5 - (-1) \cdot (-3)) - \hat{j}(1 \cdot 5 - (-1) \cdot 2) + \hat{k}(1 \cdot (-3) - 4 \cdot 2) \)

= \( \hat{i}(20 - 3) - \hat{j}(5 + 2) + \hat{k}(-3 - 8) \)

= \( \hat{i} \cdot 17 - \hat{j} \cdot 7 - \hat{k} \cdot 11 \)

= \( 17\hat{i} - 7\hat{j} - 11\hat{k} \)

Step 2: Compute the dot product \( \vec{p} \cdot (\vec{q} \times \vec{r}) \).

\( \vec{p} \cdot (\vec{q} \times \vec{r}) = (3\hat{i} - \hat{j} + 2\hat{k}) \cdot (17\hat{i} - 7\hat{j} - 11\hat{k}) \)

= \( 3 \cdot 17 + (-1) \cdot (-7) + 2 \cdot (-11) \)

= \( 51 + 7 - 22 \)

= \( 36 \)

The final answer is \( 36 \).

Answer 2

To solve the problem, we need to find \( \vec{p} \cdot (\vec{q} \times \vec{r}) \). This involves two steps: first, compute the cross product \(\vec{q} \times \vec{r}\), and then calculate the dot product of \(\vec{p}\) with this result.

1. Calculate the cross product \(\vec{q} \times \vec{r}\):
   Given \(\vec{q} = \hat{i} + 4\hat{j} - \hat{k}\) and \(\vec{r} = 2\hat{i} - 3\hat{j} + 5\hat{k}\), the cross product is:
   \(\vec{q} \times \vec{r} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 4 & -1 \\ 2 & -3 & 5 \end{vmatrix}\)
   This determinant can be expanded as follows:
   \(\vec{q} \times \vec{r} = \hat{i}(4 \cdot 5 - (-1) \cdot (-3)) - \hat{j}(1 \cdot 5 - (-1) \cdot 2) + \hat{k}(1 \cdot -3 - 4 \cdot 2)\)
   \(= \hat{i}(20 - 3) - \hat{j}(5 + 2) + \hat{k}(-3 - 8)\)
   \(= 17\hat{i} - 7\hat{j} - 11\hat{k}\)
2. Calculate the dot product \(\vec{p} \cdot (\vec{q} \times \vec{r})\):
   Given \(\vec{p} = 3\hat{i} - \hat{j} + 2\hat{k}\), we have:
   \(\vec{p} \cdot (\vec{q} \times \vec{r}) = (3\hat{i} - \hat{j} + 2\hat{k}) \cdot (17\hat{i} - 7\hat{j} - 11\hat{k})\)
   Compute the dot product:
   \(= 3 \cdot 17 + (-1) \cdot (-7) + 2 \cdot (-11)\)
   \(= 51 + 7 - 22\)
   \(= 36\)

The result is \( \vec{p} \cdot (\vec{q} \times \vec{r}) = 36\).