If bar{a}, bar{b}, bar{c} are nonzero vectors | vedclass

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

Question

(C) The volume of a parallelepiped with coterminus edges $\bar{a}, \bar{b}, \bar{c}$ is given by the scalar triple product $[\bar{a} \bar{b} \bar{c}]

Vedclass · Accepted Answer

$14$ cubic units

Vedclass · Answer

(C) The volume of a parallelepiped with coterminus edges $\bar{a}, \bar{b}, \bar{c}$ is given by the scalar triple product $[\bar{a} \bar{b} \bar{c}] = |\bar{a} \cdot (\bar{b} 	imes \bar{c})| = 7$. We need to find the volume of a parallelepiped with edges $\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}$. This volume is given by the scalar triple product $[\bar{a}+\bar{b} \quad \bar{b}+\bar{c} \quad \bar{c}+\bar{a}]$. Using the property of the scalar triple product,we have: $[\bar{a}+\bar{b} \quad \bar{b}+\bar{c} \quad \bar{c}+\bar{a}] = (\bar{a}+\bar{b}) \cdot ((\bar{b}+\bar{c}) 	imes (\bar{c}+\bar{a}))$. Expanding the cross product: $(\bar{b}+\bar{c}) 	imes (\bar{c}+\bar{a}) = \bar{b} 	imes \bar{c} + \bar{b} 	imes \bar{a} + \bar{c} 	imes \bar{c} + \bar{c} 	imes \bar{a} = \bar{b} 	imes \bar{c} + \bar{b} 	imes \bar{a} + \bar{c} 	imes \bar{a}$ (since $\bar{c} 	imes \bar{c} = 0$). Now,taking the dot product with $(\bar{a}+\bar{b})$: $(\bar{a}+\bar{b}) \cdot (\bar{b} 	imes \bar{c} + \bar{b} 	imes \bar{a} + \bar{c} 	imes \bar{a}) = \bar{a} \cdot (\bar{b} 	imes \bar{c}) + \bar{a} \cdot (\bar{b} 	imes \bar{a}) + \bar{a} \cdot (\bar{c} 	imes \bar{a}) + \bar{b} \cdot (\bar{b} 	imes \bar{c}) + \bar{b} \cdot (\bar{b} 	imes \bar{a}) + \bar{b} \cdot (\bar{c} 	imes \bar{a})$. Since the scalar triple product is zero if any two vectors are the same,we have: $= [\bar{a} \bar{b} \bar{c}] + 0 + 0 + 0 + 0 + [\bar{b} \bar{c} \bar{a}] = [\bar{a} \bar{b} \bar{c}] + [\bar{a} \bar{b} \bar{c}] = 2[\bar{a} \bar{b} \bar{c}]$. Given $[\bar{a} \bar{b} \bar{c}] = 7$,the volume is $2 	imes 7 = 14$ cubic units.

If $\bar{a}, \bar{b}, \bar{c}$ are nonzero vectors along the coterminus edges of a parallelepiped with volume $7$ cubic units,then the volume of a parallelepiped with $\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}$ as the coterminus edges is

Similar Questions

If the scalar triple product of the vectors $-3 \hat{i}+7 \hat{j}-3 \hat{k}$,$3 \hat{i}-7 \hat{j}+\lambda \hat{k}$ and $7 \hat{i}-5 \hat{j}-3 \hat{k}$ is $272$,then $\lambda = \ldots$

If $\vec{a}, \vec{b}, \vec{c}$ are three non-coplanar vectors and $\vec{r}$ is any vector,then $[\vec{b} \, \vec{c} \, \vec{r}] \vec{a} + [\vec{c} \, \vec{a} \, \vec{r}] \vec{b} + [\vec{a} \, \vec{b} \, \vec{r}] \vec{c} = \dots$

Let $a=2 \hat{i}+\hat{j}-3 \hat{k}$ and $b=\hat{i}+3 \hat{j}+2 \hat{k}$. Then the volume of the parallelopiped having coterminous edges as $a, b$ and $c$, where $c$ is the vector perpendicular to the plane of $a, b$ and $|c|=2$ is

Let $\vec{b} = -\hat{i} + 4\hat{j} + 6\hat{k}$ and $\vec{c} = 2\hat{i} - 7\hat{j} - 10\hat{k}$. If $\vec{a}$ is a unit vector and the scalar triple product $[\vec{a} \ \vec{b} \ \vec{c}]$ has the greatest value,then $\vec{a}$ is equal to:

If $a, b, c$ are non-coplanar vectors and $\lambda$ is a real number,then $[\lambda(a + b), \lambda^2 b, \lambda c] = [a, b + c, b]$ for

Vedclass Test Series

Exam Paper Generator

Online Exam Module