(overrightarrow{a}+2 overrightarrow{b}-overri | vedclass

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

Question

(C) Let the expression be $E = (\overrightarrow{a}+2 \overrightarrow{b}-\overrightarrow{c}) \cdot ((\overrightarrow{a}-\overrightarrow{b}) 	imes (\ov

Vedclass · Accepted Answer

$3[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]$

Vedclass · Answer

(C) Let the expression be $E = (\overrightarrow{a}+2 \overrightarrow{b}-\overrightarrow{c}) \cdot ((\overrightarrow{a}-\overrightarrow{b}) 	imes (\overrightarrow{a}-\overrightarrow{b}-\overrightarrow{c}))$.First,simplify the cross product term: $(\overrightarrow{a}-\overrightarrow{b}) 	imes (\overrightarrow{a}-\overrightarrow{b}-\overrightarrow{c})$.$= (\overrightarrow{a} 	imes \overrightarrow{a}) - (\overrightarrow{a} 	imes \overrightarrow{b}) - (\overrightarrow{a} 	imes \overrightarrow{c}) - (\overrightarrow{b} 	imes \overrightarrow{a}) + (\overrightarrow{b} 	imes \overrightarrow{b}) + (\overrightarrow{b} 	imes \overrightarrow{c})$.Since $\overrightarrow{a} 	imes \overrightarrow{a} = \overrightarrow{0}$ and $\overrightarrow{b} 	imes \overrightarrow{b} = \overrightarrow{0}$,and using $\overrightarrow{b} 	imes \overrightarrow{a} = -(\overrightarrow{a} 	imes \overrightarrow{b})$,we get:$= \overrightarrow{0} - (\overrightarrow{a} 	imes \overrightarrow{b}) - (\overrightarrow{a} 	imes \overrightarrow{c}) + (\overrightarrow{a} 	imes \overrightarrow{b}) + \overrightarrow{0} + (\overrightarrow{b} 	imes \overrightarrow{c}) = (\overrightarrow{b} 	imes \overrightarrow{c}) - (\overrightarrow{a} 	imes \overrightarrow{c}) = (\overrightarrow{b} 	imes \overrightarrow{c}) + (\overrightarrow{c} 	imes \overrightarrow{a})$.Now,$E = (\overrightarrow{a}+2 \overrightarrow{b}-\overrightarrow{c}) \cdot ((\overrightarrow{b} 	imes \overrightarrow{c}) + (\overrightarrow{c} 	imes \overrightarrow{a}))$.Using the definition of scalar triple product $[\overrightarrow{u} \overrightarrow{v} \overrightarrow{w}] = \overrightarrow{u} \cdot (\overrightarrow{v} 	imes \overrightarrow{w})$:$E = \overrightarrow{a} \cdot (\overrightarrow{b} 	imes \overrightarrow{c}) + \overrightarrow{a} \cdot (\overrightarrow{c} 	imes \overrightarrow{a}) + 2\overrightarrow{b} \cdot (\overrightarrow{b} 	imes \overrightarrow{c}) + 2\overrightarrow{b} \cdot (\overrightarrow{c} 	imes \overrightarrow{a}) - \overrightarrow{c} \cdot (\overrightarrow{b} 	imes \overrightarrow{c}) - \overrightarrow{c} \cdot (\overrightarrow{c} 	imes \overrightarrow{a})$.Note that any scalar triple product with repeated vectors is $0$ (e.g.,$\overrightarrow{a} \cdot (\overrightarrow{c} 	imes \overrightarrow{a}) = 0$).$E = [\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}] + 0 + 0 + 2[\overrightarrow{b} \overrightarrow{c} \overrightarrow{a}] - 0 - 0$.Since $[\overrightarrow{b} \overrightarrow{c} \overrightarrow{a}] = [\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]$,we have $E = [\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}] + 2[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}] = 3[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]$.

$(\overrightarrow{a}+2 \overrightarrow{b}-\overrightarrow{c}) \cdot ((\overrightarrow{a}-\overrightarrow{b}) \times (\overrightarrow{a}-\overrightarrow{b}-\overrightarrow{c}))$ is equal to

Similar Questions

Let $p, q, r$ be three non-coplanar vectors and $b = p \times q$. If $a, b, c$ denote the coterminous edges of a parallelepiped,then its height with the base having $a$ and $c$ is

Let $a = i - j$,$b = j - k$,$c = k - i$. If $\hat{d}$ is a unit vector such that $a \cdot \hat{d} = 0$ and $[b, c, \hat{d}] = 0$,then $\hat{d}$ is equal to

If $u, v$ and $w$ are three non-coplanar vectors,then $(u + v - w) \cdot [(u - v) \times (v - w)]$ equals

The vectors $2 \hat{i}-3 \hat{j}+\hat{k}, \hat{i}-2 \hat{j}+3 \hat{k}$ and $3 \hat{i}+\hat{j}-2 \hat{k}$

Let $a=p(\hat{i}+\hat{j}+\hat{k})$,$b=\hat{i}+\hat{j}-2\hat{k}$,and $c=2\hat{i}-\hat{j}+2\hat{k}$ be three vectors. If the value of $[abc]$ is not more than $15$ and not less than $-5$,then $p$ lies in the interval:

Vedclass Test Series

Exam Paper Generator

Online Exam Module