For any non-zero vectors a, b, c,a cdot[(b+c) | vedclass

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

Question

(A) We have the expression $a \cdot[(b+c) 	imes(a+b+c)]$. Using the distributive property of the cross product,we expand the term inside the bracket:

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(A) We have the expression $a \cdot[(b+c) 	imes(a+b+c)]$. Using the distributive property of the cross product,we expand the term inside the bracket: $(b+c) 	imes(a+b+c) = (b 	imes a) + (b 	imes b) + (b 	imes c) + (c 	imes a) + (c 	imes b) + (c 	imes c)$. Since the cross product of any vector with itself is zero ($b 	imes b = 0$ and $c 	imes c = 0$) and $c 	imes b = -(b 	imes c)$,the expression simplifies to: $(b 	imes a) + (b 	imes c) + (c 	imes a) - (b 	imes c) = (b 	imes a) + (c 	imes a)$. Now,taking the dot product with $a$: $a \cdot [(b 	imes a) + (c 	imes a)] = a \cdot (b 	imes a) + a \cdot (c 	imes a)$. By the definition of the scalar triple product,$a \cdot (b 	imes a) = [a \ b \ a]$ and $a \cdot (c 	imes a) = [a \ c \ a]$. Since any scalar triple product with two identical vectors is zero,we have $[a \ b \ a] = 0$ and $[a \ c \ a] = 0$. Therefore,the final result is $0 + 0 = 0$.

For any non-zero vectors $a, b, c$,$a \cdot[(b+c) \times(a+b+c)] = \ldots .$

Similar Questions

For what value of $a$ is the volume of the parallelepiped formed by the vectors $i + aj + k$,$j + ak$,and $ai + k$ minimum?

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

If the volume of a parallelepiped with coterminous edges $4 \hat{i} + 5 \hat{j} + \hat{k}$, $-\hat{j} + \hat{k}$, and $3 \hat{i} + 9 \hat{j} + p \hat{k}$ is $34$ cubic units, then $p$ is equal to:

If $\vec{w} = \alpha (\vec{a} \times \vec{b}) + \beta (\vec{b} \times \vec{c}) + \gamma (\vec{c} \times \vec{a})$,$[\vec{a}, \vec{b}, \vec{c}] = 2$ and $\vec{w} \cdot (\vec{a} + \vec{b} + \vec{c}) = 8$,then $\alpha + \beta + \gamma =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module