Scalar Triple Product Calculation
The scalar triple product $\vec{a} \cdot (\vec{b} \times \vec{c})$ is calculated using the determinant of the matrix formed by the components of the vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$.
Given Vectors
- $\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$
- $\vec{b} = \hat{i} + 2\hat{j} - 3\hat{k}$
- $\vec{c} = 3\hat{i} + 4\hat{j} - \hat{k}$
Determinant Method
The scalar triple product equals the determinant:
$ \vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} 2 & -3 & 4 \\ 1 & 2 & -3 \\ 3 & 4 & -1 \end{vmatrix} $Evaluating the Determinant
Expand the determinant along the first row:
$ = 2 \begin{vmatrix} 2 & -3 \\ 4 & -1 \end{vmatrix} - (-3) \begin{vmatrix} 1 & -3 \\ 3 & -1 \end{vmatrix} + 4 \begin{vmatrix} 1 & 2 \\ 3 & 4 \end{vmatrix} $Calculate the 2x2 determinants:
- $ \begin{vmatrix} 2 & -3 \\ 4 & -1 \end{vmatrix} = (2 \times -1) - (-3 \times 4) = -2 - (-12) = -2 + 12 = 10 $
- $ \begin{vmatrix} 1 & -3 \\ 3 & -1 \end{vmatrix} = (1 \times -1) - (-3 \times 3) = -1 - (-9) = -1 + 9 = 8 $
- $ \begin{vmatrix} 1 & 2 \\ 3 & 4 \end{vmatrix} = (1 \times 4) - (2 \times 3) = 4 - 6 = -2 $
Substitute these values back into the expansion:
$ = 2(10) + 3(8) + 4(-2) $ $ = 20 + 24 - 8 $ $ = 44 - 8 $ $ = 36 $Result
The scalar triple product $\vec{a} \cdot (\vec{b} \times \vec{c})$ is 36.
Important Questions from Vector Algebra
What is the length of projection of the vector \(\rm \hat{i}+2 \hat{j}+3 \hat{k}\) on the vector \(\rm2 \hat{i}+3 \hat{j}-2 \hat{k}\) ?
Consider the following in respect of the vectors \(\rm \vec{a}=(0,1,1)\) and \(\rm \vec{b}=(1,0,1) \) :
1. The number of unit vectors perpendicular to both \(\rm \vec{a}\) and \(\rm \vec{b}\) is only one.
2. The angle between the vectors is \(\frac{\pi}{3}\).
Which of the statements given above is/are correct?
Consider the following points :
1. (-1, -3, 1)
2. (-1, 3, 2)
3. (-2, 5, 3)
Which of the above points lie on the line joining A and B ?
What is the magnitude of \(\overrightarrow{A B}\) ?
If \({\rm{\vec d}} = {\rm{x\hat i}} + {\rm{y\hat j}} + {\rm{z\hat k}}\) , then which of the following equations is/are correct?
1. y – x = 4
2. 2z – 3 = 0
Select the correct answer using the code given below:
