If 3 hat{j},4 hat{k} and 3 hat{j}+4 hat{k} ar | vedclass

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

Question

(D) Let the position vectors of vertices $A$,$B$,and $C$ be $\vec{a} = 3 \hat{j}$,$\vec{b} = 4 \hat{k}$,and $\vec{c} = 3 \hat{j} + 4 \hat{k}$ respecti

Vedclass · Accepted Answer

$\frac{5}{3} \hat{j}+4 \hat{k}$

Vedclass · Answer

(D) Let the position vectors of vertices $A$,$B$,and $C$ be $\vec{a} = 3 \hat{j}$,$\vec{b} = 4 \hat{k}$,and $\vec{c} = 3 \hat{j} + 4 \hat{k}$ respectively.According to the Angle Bisector Theorem,the bisector of $\angle A$ divides the opposite side $BC$ in the ratio of the lengths of the sides adjacent to the angle,i.e.,$AB : AC$.First,calculate the lengths of sides $AB$ and $AC$:$AB = |\vec{b} - \vec{a}| = |4 \hat{k} - 3 \hat{j}| = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = 5$.$AC = |\vec{c} - \vec{a}| = |(3 \hat{j} + 4 \hat{k}) - 3 \hat{j}| = |4 \hat{k}| = 4$.Thus,the bisector of $\angle A$ divides $BC$ in the ratio $m : n = 5 : 4$.The position vector $\vec{d}$ of the point $D$ on $BC$ is given by the section formula:$\vec{d} = \frac{m \vec{c} + n \vec{b}}{m + n} = \frac{5(3 \hat{j} + 4 \hat{k}) + 4(4 \hat{k})}{5 + 4}$.$\vec{d} = \frac{15 \hat{j} + 20 \hat{k} + 16 \hat{k}}{9} = \frac{15 \hat{j} + 36 \hat{k}}{9}$.$\vec{d} = \frac{15}{9} \hat{j} + \frac{36}{9} \hat{k} = \frac{5}{3} \hat{j} + 4 \hat{k}$.

If $3 \hat{j}$,$4 \hat{k}$ and $3 \hat{j}+4 \hat{k}$ are the position vectors of the vertices $A$,$B$,and $C$ respectively of $\triangle ABC$,then the position vector of the point in which the bisector of $\angle A$ meets $BC$ is

Similar Questions

If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}|=2$,$|\vec{b}|=3$ and $\vec{a}+t \vec{b}$ and $\vec{a}-t \vec{b}$ are perpendicular,where $t$ is a positive scalar,then

The vectors $2\,i + 3\,j - 4\,k$ and $a\,i + b\,j + c\,k$ are perpendicular,when

If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}| = 2$ and $|\vec{b}| = 3$, then the maximum value of $3 |(3\vec{a} + 2\vec{b})| + 4 |(3\vec{a} - 2\vec{b})|$ is:

If the position vectors of the points $A$ and $B$ are $2\,i + 3\,j - k$ and $-2\,i + 3\,j + 4\,k$,then the line $AB$ is parallel to

If $\triangle ABC$ is right-angled at $A$,where $A \equiv (4, 2, x)$,$B \equiv (3, 1, 8)$,and $C \equiv (2, -1, 2)$,then the value of $x$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module