If |vec{a}| = 4, |vec{b}| = 3 and vec{a} cdot | vedclass

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

Question

(B) The scalar triple product is defined as $[\vec{a} \quad \vec{a} + \vec{b} \quad \vec{a} 	imes \vec{b}] = \vec{a} \cdot ((\vec{a} + \vec{b}) 	ime

Vedclass · Accepted Answer

$80$

Vedclass · Answer

(B) The scalar triple product is defined as $[\vec{a} \quad \vec{a} + \vec{b} \quad \vec{a} 	imes \vec{b}] = \vec{a} \cdot ((\vec{a} + \vec{b}) 	imes (\vec{a} 	imes \vec{b}))$.Using the vector triple product identity $\vec{u} 	imes (\vec{v} 	imes \vec{w}) = (\vec{u} \cdot \vec{w})\vec{v} - (\vec{u} \cdot \vec{v})\vec{w}$, we expand $(\vec{a} + \vec{b}) 	imes (\vec{a} 	imes \vec{b})$:$(\vec{a} + \vec{b}) 	imes (\vec{a} 	imes \vec{b}) = \vec{a} 	imes (\vec{a} 	imes \vec{b}) + \vec{b} 	imes (\vec{a} 	imes \vec{b})$$= ((\vec{a} \cdot \vec{b})\vec{a} - (\vec{a} \cdot \vec{a})\vec{b}) + ((\vec{b} \cdot \vec{b})\vec{a} - (\vec{b} \cdot \vec{a})\vec{b})$$= (\vec{a} \cdot \vec{b})\vec{a} - |\vec{a}|^2 \vec{b} + |\vec{b}|^2 \vec{a} - (\vec{a} \cdot \vec{b})\vec{b}$Now, take the dot product with $\vec{a}$:$[\vec{a} \quad \vec{a} + \vec{b} \quad \vec{a} 	imes \vec{b}] = \vec{a} \cdot [(\vec{a} \cdot \vec{b})\vec{a} - |\vec{a}|^2 \vec{b} + |\vec{b}|^2 \vec{a} - (\vec{a} \cdot \vec{b})\vec{b}]$$= (\vec{a} \cdot \vec{b})(\vec{a} \cdot \vec{a}) - |\vec{a}|^2 (\vec{a} \cdot \vec{b}) + |\vec{b}|^2 (\vec{a} \cdot \vec{a}) - (\vec{a} \cdot \vec{b})(\vec{a} \cdot \vec{b})$$= (8)(16) - (16)(8) + (9)(16) - (8)(8)$$= 0 + 144 - 64 = 80$.

If $|\vec{a}| = 4, |\vec{b}| = 3$ and $\vec{a} \cdot \vec{b} = 8$, then the scalar triple product $[\vec{a} \quad \vec{a} + \vec{b} \quad \vec{a} \times \vec{b}]$ is equal to:

Similar Questions

If $A(-1, 2, 3)$,$B(3, -2, 1)$,$C(2, 1, 3)$ and $D(-1, -2, 4)$ are the vertices of a tetrahedron,then its volume is

$a \cdot (a \times b) = $

If $a = i - j + k$,$b = i + 2j - k$ and $c = 3i + pj + 5k$ are coplanar,then the value of $p$ is:

If three vectors $\vec{a} = 12\hat{i} + 4\hat{j} + 3\hat{k}$,$\vec{b} = 8\hat{i} - 12\hat{j} - 9\hat{k}$,and $\vec{c} = 33\hat{i} - 4\hat{j} - 24\hat{k}$ represent the coterminous edges of a parallelepiped,then its volume is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module