If bar{a}=a_1 hat{i}+a_2 hat{j}+a_3 hat{k}, b | vedclass

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

Question

(B) First,note that the determinant on the right side is equal to the scalar triple product $[\bar{a} \quad \bar{b} \quad \bar{c}]$.$\begin{vmatrix} a

Vedclass · Accepted Answer

$28$

Vedclass · Answer

(B) First,note that the determinant on the right side is equal to the scalar triple product $[\bar{a} \quad \bar{b} \quad \bar{c}]$.$\begin{vmatrix} a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \end{vmatrix} = [\bar{a} \quad \bar{b} \quad \bar{c}]$.Now,calculate the scalar triple product on the left side:$[3 \bar{a}+\bar{b} \quad 3 \bar{b}+\bar{c} \quad 3 \bar{c}+\bar{a}] = (3 \bar{a}+\bar{b}) \cdot ((3 \bar{b}+\bar{c}) 	imes (3 \bar{c}+\bar{a}))$.Expanding the cross product:$(3 \bar{b}+\bar{c}) 	imes (3 \bar{c}+\bar{a}) = 9(\bar{b} 	imes \bar{c}) + 3(\bar{b} 	imes \bar{a}) + 3(\bar{c} 	imes \bar{c}) + (\bar{c} 	imes \bar{a}) = 9(\bar{b} 	imes \bar{c}) + 3(\bar{b} 	imes \bar{a}) + (\bar{c} 	imes \bar{a})$.Now take the dot product with $(3 \bar{a}+\bar{b})$:$= (3 \bar{a}+\bar{b}) \cdot (9(\bar{b} 	imes \bar{c}) + 3(\bar{b} 	imes \bar{a}) + (\bar{c} 	imes \bar{a}))$$= 27[\bar{a} \quad \bar{b} \quad \bar{c}] + 9[\bar{a} \quad \bar{b} \quad \bar{a}] + 3[\bar{a} \quad \bar{c} \quad \bar{a}] + 9[\bar{b} \quad \bar{b} \quad \bar{c}] + 3[\bar{b} \quad \bar{b} \quad \bar{a}] + [\bar{b} \quad \bar{c} \quad \bar{a}]$.Since any scalar triple product with two identical vectors is $0$,this simplifies to:$= 27[\bar{a} \quad \bar{b} \quad \bar{c}] + [\bar{b} \quad \bar{c} \quad \bar{a}] = 27[\bar{a} \quad \bar{b} \quad \bar{c}] + [\bar{a} \quad \bar{b} \quad \bar{c}] = 28[\bar{a} \quad \bar{b} \quad \bar{c}]$.Comparing this with $\lambda [\bar{a} \quad \bar{b} \quad \bar{c}]$,we get $\lambda = 28$.

Similar Questions

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors,then $\frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{\vec{c} \cdot (\vec{a} \times \vec{b})} + \frac{\vec{b} \cdot (\vec{a} \times \vec{c})}{\vec{c} \cdot (\vec{a} \times \vec{b})} = \dots$

If $a$ and $b$ are parallel vectors,then $[a \ c \ b] = $

If $p = \frac{b \times c}{[a, b, c]}, q = \frac{c \times a}{[a, b, c]}, r = \frac{a \times b}{[a, b, c]}$,where $a, b, c$ are three non-coplanar vectors,then the value of $(a + b + c) \cdot (p + q + r)$ is given by

Which of the following is not equal to $w \cdot(u \times v)$ ?

Let $\bar{a}$ and $\bar{c}$ be unit vectors at an angle $\frac{\pi}{3}$ with each other. If $(\bar{a} \times(\bar{b} \times \bar{c})) \cdot(\bar{a} \times \bar{c})=5$,then $\left[\begin{array}{lll}\bar{a} & \bar{b} & \bar{c}\end{array}\right]=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module