The area of the triangle whose vertices are A | vedclass

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

Question

(B) The area of a triangle with vertices $A$,$B$,and $C$ is given by the formula $	ext{Area} = \frac{1}{2} |\vec{AB} 	imes \vec{AC}|$.Given vertices

Vedclass · Accepted Answer

$\frac{\sqrt{21}}{2}$

Vedclass · Answer

(B) The area of a triangle with vertices $A$,$B$,and $C$ is given by the formula $	ext{Area} = \frac{1}{2} |\vec{AB} 	imes \vec{AC}|$.Given vertices are $A(1, 1, 1)$,$B(1, 2, 3)$,and $C(2, 3, 1)$.First,find the vectors $\vec{AB}$ and $\vec{AC}$:$\vec{AB} = (1-1)\hat{i} + (2-1)\hat{j} + (3-1)\hat{k} = 0\hat{i} + 1\hat{j} + 2\hat{k} = \hat{j} + 2\hat{k}$.$\vec{AC} = (2-1)\hat{i} + (3-1)\hat{j} + (1-1)\hat{k} = 1\hat{i} + 2\hat{j} + 0\hat{k} = \hat{i} + 2\hat{j}$.Now,calculate the cross product $\vec{AB} 	imes \vec{AC}$:$\vec{AB} 	imes \vec{AC} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 0 & 1 & 2 \ 1 & 2 & 0 \end{vmatrix} = \hat{i}(0 - 4) - \hat{j}(0 - 2) + \hat{k}(0 - 1) = -4\hat{i} + 2\hat{j} - 1\hat{k}$.The magnitude of the cross product is $|\vec{AB} 	imes \vec{AC}| = \sqrt{(-4)^2 + 2^2 + (-1)^2} = \sqrt{16 + 4 + 1} = \sqrt{21}$.Therefore,the area of the triangle is $\frac{1}{2} \sqrt{21}$ sq. units.

The area of the triangle whose vertices are $A(1, 1, 1)$,$B(1, 2, 3)$,and $C(2, 3, 1)$ is . . . . . . sq. units.

Similar Questions

If $|\vec{a}| = 4$,$|\vec{b}| = 2$ and the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{6}$,then $|\vec{a} \times \vec{b}|^2 = \dots$

Let $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ be the position vectors of the vertices $A, B, C$ respectively of $\triangle ABC$. The vector area of $\triangle ABC$ is:

Show that $(\vec{a}-\vec{b}) \times(\vec{a}+\vec{b})=2(\vec{a} \times \vec{b})$.

The adjacent sides of a parallelogram are $\vec{a} = 3\hat{i} + \hat{j} + 4\hat{k}$ and $\vec{b} = \hat{i} - \hat{j} + \hat{k}$. Then,the area of the parallelogram is . . . . . . sq. units.

If $a = 2i + 2j - k$ and $b = 6i - 3j + 2k,$ then the value of $a \times b$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module