If vec{a}, vec{b} and vec{c} are three non-co | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let the scalar triple product be $[\vec{a} \vec{b} \vec{c}] = \vec{a} \cdot (\vec{b} 	imes \vec{c})$. Since $\vec{a}, \vec{b}, \vec{c}$ are non-c

Vedclass · Accepted Answer

$3$

Vedclass · Answer

(B) Let the scalar triple product be $[\vec{a} \vec{b} \vec{c}] = \vec{a} \cdot (\vec{b} 	imes \vec{c})$. Since $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar,$[\vec{a} \vec{b} \vec{c}] 
eq 0$. Given $\vec{p} = \frac{\vec{b} 	imes \vec{c}}{[\vec{a} \vec{b} \vec{c}]}, \vec{q} = \frac{\vec{c} 	imes \vec{a}}{[\vec{a} \vec{b} \vec{c}]}, \vec{r} = \frac{\vec{a} 	imes \vec{b}}{[\vec{a} \vec{b} \vec{c}]}$. We need to evaluate $S = (\vec{a}+\vec{b}) \cdot \vec{p} + (\vec{b}+\vec{c}) \cdot \vec{q} + (\vec{c}+\vec{a}) \cdot \vec{r}$. Substituting the expressions: $S = \frac{(\vec{a}+\vec{b}) \cdot (\vec{b} 	imes \vec{c})}{[\vec{a} \vec{b} \vec{c}]} + \frac{(\vec{b}+\vec{c}) \cdot (\vec{c} 	imes \vec{a})}{[\vec{a} \vec{b} \vec{c}]} + \frac{(\vec{c}+\vec{a}) \cdot (\vec{a} 	imes \vec{b})}{[\vec{a} \vec{b} \vec{c}]}$. Using the property $\vec{a} \cdot (\vec{b} 	imes \vec{c}) = [\vec{a} \vec{b} \vec{c}]$ and $\vec{b} \cdot (\vec{b} 	imes \vec{c}) = 0$: $S = \frac{[\vec{a} \vec{b} \vec{c}] + 0}{[\vec{a} \vec{b} \vec{c}]} + \frac{[\vec{b} \vec{c} \vec{a}] + 0}{[\vec{a} \vec{b} \vec{c}]} + \frac{[\vec{c} \vec{a} \vec{b}] + 0}{[\vec{a} \vec{b} \vec{c}]}$. Since $[\vec{a} \vec{b} \vec{c}] = [\vec{b} \vec{c} \vec{a}] = [\vec{c} \vec{a} \vec{b}]$,we get: $S = 1 + 1 + 1 = 3$.

Similar Questions

The altitude of the parallelepiped,whose coterminus edges are the vectors $\bar{a}=\hat{i}+\hat{j}+\hat{k}$,$\bar{b}=2\hat{i}+4\hat{j}-\hat{k}$,and $\bar{c}=\hat{i}+\hat{j}+3\hat{k}$,where $\bar{a}$ and $\bar{b}$ are the sides of the base of the parallelepiped,is:

For non-zero vectors $\vec{a}, \vec{b}, \vec{c}$,the condition $|(\vec{a} \times \vec{b}) \cdot \vec{c}| = |\vec{a}||\vec{b}||\vec{c}|$ holds if and only if:

The number of integral values of $p$ for which the vectors $(p+1) \hat{i} - 3 \hat{j} + p \hat{k}$,$p \hat{i} + (p+1) \hat{j} - 3 \hat{k}$,and $-3 \hat{i} + p \hat{j} + (p+1) \hat{k}$ are linearly dependent is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module