The scalar overline{a} cdot [(overline{b} + o | vedclass

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

Question

(A) Given expression: $\overline{a} \cdot [(\overline{b} + \overline{c}) 	imes (\overline{a} + \overline{b} + \overline{c})]$ Using the distributive

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(A) Given expression: $\overline{a} \cdot [(\overline{b} + \overline{c}) 	imes (\overline{a} + \overline{b} + \overline{c})]$ Using the distributive property of the cross product: $(\overline{b} + \overline{c}) 	imes (\overline{a} + \overline{b} + \overline{c}) = \overline{b} 	imes \overline{a} + \overline{b} 	imes \overline{b} + \overline{b} 	imes \overline{c} + \overline{c} 	imes \overline{a} + \overline{c} 	imes \overline{b} + \overline{c} 	imes \overline{c}$ Since $\overline{x} 	imes \overline{x} = 0$ and $\overline{c} 	imes \overline{b} = -(\overline{b} 	imes \overline{c})$,we have: $= \overline{b} 	imes \overline{a} + 0 + \overline{b} 	imes \overline{c} + \overline{c} 	imes \overline{a} - (\overline{b} 	imes \overline{c}) + 0 = \overline{b} 	imes \overline{a} + \overline{c} 	imes \overline{a}$ Now,taking the dot product with $\overline{a}$: $\overline{a} \cdot (\overline{b} 	imes \overline{a} + \overline{c} 	imes \overline{a}) = \overline{a} \cdot (\overline{b} 	imes \overline{a}) + \overline{a} \cdot (\overline{c} 	imes \overline{a})$ $= [\overline{a} \overline{b} \overline{a}] + [\overline{a} \overline{c} \overline{a}]$ Since the scalar triple product is zero if any two vectors are identical: $= 0 + 0 = 0$

The scalar $\overline{a} \cdot [(\overline{b} + \overline{c}) \times (\overline{a} + \overline{b} + \overline{c})]$ equals

Similar Questions

The volume of the parallelopiped whose coterminous edges are $\hat{j}+\hat{k}$, $\hat{i}+\hat{k}$, and $\hat{i}+\hat{j}$ is

Let the volume of the tetrahedron with vertices $\hat{i}-\hat{j}-2\hat{k}$,$-2\hat{i}+\hat{j}-2\hat{k}$,$-\hat{i}-2\hat{j}+\hat{k}$,and $2\hat{i}+2\hat{j}+a\hat{k}$ be $\frac{20}{3}$. Then the integral value of $a$ is

The vector $c \cdot (b+c) \times (a+b+c)$ is equal to

If the points with position vectors $\hat{i}-2 \hat{j}+3 \hat{k}$,$2 \hat{i}+3 \hat{j}-4 \hat{k}$,$-3 \hat{i}+\hat{j}-5 \hat{k}$,and $a \hat{i}-2 \hat{j}+4 \hat{k}$ are coplanar,then $a=$

If $[\bar{a} \quad \bar{b} \quad \bar{c}]=4$,then the volume of the parallelepiped with coterminous edges $\bar{a}+2 \bar{b}$,$\bar{b}+2 \bar{c}$,and $\bar{c}+2 \bar{a}$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module