If vec{a}=2 hat{i}+hat{j}+3 hat{k},vec{b}=hat | vedclass

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

Question

(B) Given vectors are $\vec{a}=2 \hat{i}+\hat{j}+3 \hat{k}$,$\vec{b}=\hat{i}+3 \hat{j}-\hat{k}$,and $\vec{c}=3 \hat{i}-\hat{j}-2 \hat{k}$.First,we cal

Vedclass · Accepted Answer

$2025$

Vedclass · Answer

(B) Given vectors are $\vec{a}=2 \hat{i}+\hat{j}+3 \hat{k}$,$\vec{b}=\hat{i}+3 \hat{j}-\hat{k}$,and $\vec{c}=3 \hat{i}-\hat{j}-2 \hat{k}$.First,we calculate the dot products:$\vec{a} \cdot \vec{a} = (2)^2 + (1)^2 + (3)^2 = 4 + 1 + 9 = 14$$\vec{b} \cdot \vec{b} = (1)^2 + (3)^2 + (-1)^2 = 1 + 9 + 1 = 11$$\vec{c} \cdot \vec{c} = (3)^2 + (-1)^2 + (-2)^2 = 9 + 1 + 4 = 14$$\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} = (2)(1) + (1)(3) + (3)(-1) = 2 + 3 - 3 = 2$$\vec{a} \cdot \vec{c} = \vec{c} \cdot \vec{a} = (2)(3) + (1)(-1) + (3)(-2) = 6 - 1 - 6 = -1$$\vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{b} = (1)(3) + (3)(-1) + (-1)(-2) = 3 - 3 + 2 = 2$Now,substitute these values into the determinant:$\Delta = \left|\begin{array}{ccc} 14 & 2 & -1 \ 2 & 11 & 2 \ -1 & 2 & 14 \end{array}ight|$Expanding along the first row:$\Delta = 14(11 	imes 14 - 2 	imes 2) - 2(2 	imes 14 - 2 	imes (-1)) - 1(2 	imes 2 - 11 	imes (-1))$$\Delta = 14(154 - 4) - 2(28 + 2) - 1(4 + 11)$$\Delta = 14(150) - 2(30) - 1(15)$$\Delta = 2100 - 60 - 15 = 2025$.

Similar Questions

If $\vec{OA}=6 \hat{i}+3 \hat{j}-4 \hat{k}$,$\vec{OB}=2 \hat{j}+\hat{k}$,and $\vec{OC}=5 \hat{i}-\hat{j}+2 \hat{k}$ are the coterminous edges of a parallelepiped,then the height of the parallelepiped drawn from the vertex $A$ is

If the vectors $2i - 3j$,$i + j - k$,and $3i - k$ form three concurrent edges of a parallelepiped,then the volume of the parallelepiped is

If $[\bar{a} \bar{b} \bar{c}] \neq 0$,then $\frac{[\bar{a}+\bar{b} \quad \bar{b}+\bar{c} \quad \bar{c}+\bar{a}]}{[\bar{b} \bar{c} \bar{a}]}=$

The volume of the parallelepiped whose coterminous edges are represented by the vectors $2i - 3j + 4k$,$i + 2j - 2k$,and $3i - j + k$ is ............ $cubic \ units$.

Let $a, b, c$ be distinct non-negative numbers. If the vectors $a\hat{i} + a\hat{j} + c\hat{k}$,$\hat{i} + \hat{k}$,and $c\hat{i} + c\hat{j} + b\hat{k}$ lie in a plane,then $c$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module