If 2 hat{i}+hat{j}-hat{k},hat{i}-3 hat{j}+5 h | vedclass

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

Question

(D) Given position vectors are $\overrightarrow{OA} = 2\hat{i} + \hat{j} - \hat{k}$,$\overrightarrow{OB} = \hat{i} - 3\hat{j} + 5\hat{k}$,and $\overri

Vedclass · Accepted Answer

$ABC$ is a scalene triangle

Vedclass · Answer

(D) Given position vectors are $\overrightarrow{OA} = 2\hat{i} + \hat{j} - \hat{k}$,$\overrightarrow{OB} = \hat{i} - 3\hat{j} + 5\hat{k}$,and $\overrightarrow{OC} = -3\hat{i} + 4\hat{j} + 4\hat{k}$.Calculate the vectors representing the sides:$\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA} = (1-2)\hat{i} + (-3-1)\hat{j} + (5-(-1))\hat{k} = -\hat{i} - 4\hat{j} + 6\hat{k}$.$|\overrightarrow{AB}| = \sqrt{(-1)^2 + (-4)^2 + 6^2} = \sqrt{1 + 16 + 36} = \sqrt{53}$.$\overrightarrow{BC} = \overrightarrow{OC} - \overrightarrow{OB} = (-3-1)\hat{i} + (4-(-3))\hat{j} + (4-5)\hat{k} = -4\hat{i} + 7\hat{j} - \hat{k}$.$|\overrightarrow{BC}| = \sqrt{(-4)^2 + 7^2 + (-1)^2} = \sqrt{16 + 49 + 1} = \sqrt{66}$.$\overrightarrow{CA} = \overrightarrow{OA} - \overrightarrow{OC} = (2-(-3))\hat{i} + (1-4)\hat{j} + (-1-4)\hat{k} = 5\hat{i} - 3\hat{j} - 5\hat{k}$.$|\overrightarrow{CA}| = \sqrt{5^2 + (-3)^2 + (-5)^2} = \sqrt{25 + 9 + 25} = \sqrt{59}$.Since $|\overrightarrow{AB}| 
eq |\overrightarrow{BC}| 
eq |\overrightarrow{CA}|$,all three sides have different lengths.Therefore,$	riangle ABC$ is a scalene triangle.

If $2 \hat{i}+\hat{j}-\hat{k}$,$\hat{i}-3 \hat{j}+5 \hat{k}$ and $-3 \hat{i}+4 \hat{j}+4 \hat{k}$ are the position vectors of three points $A$,$B$ and $C$ respectively,then

Similar Questions

The sum of the lengths of projections of $p \hat{i} + q \hat{j} + r \hat{k}$ on the coordinate axes, where $p = 4, q = -5, r = 7$, is: (in $\text{ units}$)

The points whose position vectors are $2\hat{i}+3\hat{j}+4\hat{k}$, $3\hat{i}+4\hat{j}+2\hat{k}$, and $4\hat{i}+2\hat{j}+3\hat{k}$ are the vertices of

The projection of the vector $i+j+k$ along the vector $j$ is

Let $ABC$ be a triangle whose circumcentre is at $P$. If the position vectors of $A, B, C$ and $P$ are $\vec{a}, \vec{b}, \vec{c}$ and $\frac{\vec{a} + \vec{b} + \vec{c}}{4}$ respectively,then the position vector of the orthocentre of this triangle is:

If the vectors $(2 \hat{\imath} - q \hat{\jmath} + 3 \hat{k})$ and $(4 \hat{\imath} - 5 \hat{\jmath} + 6 \hat{k})$ are collinear,then the value of $q$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module