Vector Triple Product Identity
The vector triple product $(a \times b) \times c$ involves the cross product of two vectors ($a \times b$) with a third vector ($c$). A standard identity simplifies this expression.
Vector Triple Product Formula
The vector triple product identity states:
$ (a \times b) \times c = (a \cdot c)b - (b \cdot c)a $
Understanding the Identity
This formula shows that the result of the vector triple product $(a \times b) \times c$ is a linear combination of vectors $b$ and $a$. The coefficients are scalar products (dot products) involving vector $c$. Specifically, it's the dot product of $a$ with $c$ times vector $b$, minus the dot product of $b$ with $c$ times vector $a$. This result is always perpendicular to $c$ and lies in the plane defined by $a$ and $b$ (unless $a$ and $b$ are parallel).
Matching with Options
We compare the derived identity with the given options:
- Option 1: $(a \cdot c)b - (a \cdot b)c$
- Option 2: $(a \cdot b)c - (a \cdot c)b$
- Option 3: $(a \cdot c)b - (b \cdot c)a$
- Option 4: $(b \cdot c)a - (a \cdot c)b$
The identity $(a \cdot c)b - (b \cdot c)a$ precisely matches Option 3.
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:
