The vertices A, B, C of triangle ABC have respectively position v… - Vidyadip

Question

The vertices A, B, C of triangle ABC have respectively position vector $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ with respect to given origin O. Show that the point D where the bisector of $\angle{\text{A}}$ meets BC has position Vector $\vec{\text{d}}=\frac{\beta\vec{\text{b}}+\gamma\vec{\text{c}}}{\beta+\gamma}$, where $\beta=\big|\vec{\text{c}}-\vec{\text{a}}\big|$ and, $\gamma=\big|\vec{\text{a}}-\vec{\text{b}}\big|$.

✓

Answer

Let the position vectors of A, B and C with respect to same origin, O be $\vec{\text{a}},\ \vec{\text{b}}\text{ and }\vec{\text{c}}$ respectively.
Let D be the point on BC where bisectors of $\angle\text{A}$ meets.
Let $\vec{\text{d}}$ be the position vector of D which divides CB internally in the ratio $\beta \text{ and } \gamma$, where
$\beta=\Big|\overrightarrow{\text{AC}}\Big|\text{ and }\gamma=\Big|\overrightarrow{\text{AB}}\Big|$
By section formula, the position vector of D is given by
$\overrightarrow{\text{OD}}=\frac{\beta\vec{\text{b}}+\gamma\vec{\text{c}}}{\beta+\gamma}$
Let $\alpha=\big|\vec{\text{b}}-\vec{\text{c}}\big|$
In center is the concurent point of angle bisectors an in center divides the line AD in the ratio $\alpha : \beta +\gamma$.
So, the position vector of in center is given as,
$\frac{\alpha\vec{\text{a}}+\Big(\frac{\beta\vec{\text{b}}+\gamma\vec{\text{c}}}{\beta+\gamma}\Big)(\beta+\gamma)}{\alpha+\beta+\gamma}=\frac{\alpha\vec{\text{a}}+\beta\vec{\text{b}}+\gamma\vec{\text{c}}}{\alpha+\beta+\gamma}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Algebra of Vectors→Algebra of Vectors — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Solve the following differential equation:
$\text{x }\frac{\text{dy}}{\text{dx}}=\text{y - x}\tan\Bigg(\frac{\text{y}}{\text{ax}}\Bigg).$

$\text{If (x}-\text{a})^2+(\text{y}-\text{b})^2=\text{c}^2,$ for some c > 0 , prove that
$\frac{\Big[1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2\Big]^{\frac{3}{2}}}{\frac{\text{d}^2\text{y}}{\text{dx}^2}}$
is a constant independent of a and b.

Solve the following initial value problems:
$\text{xe}^{\frac{\text{y}}{\text{x}}}-\text{y + x}\frac{\text{dy}}{\text{dx}}=0,\text{y(e)}=0$

Evaluate:
$\begin{vmatrix}\text{x}+\lambda&\text{x}&\text{x}\\\text{x}&\text{x}+\lambda&\text{x}\\\text{x}&\text{x}&\text{x}+\lambda\end{vmatrix}$

Show that the differential equation $y d x + x \log \left| \frac { y } { x } \right| d y - 2 x d y = 0$ is homogeneous and solve it.

Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\text{x}^{\cos\text{x}}+(\sin\text{x})^{\tan\text{x}}$

On R − {1}, a binary operation * is defined by a * b = a + b − ab. Prove that * is commutative and associative. Find the identity element for * on R − {1}. Also, prove that every element of R − {1} is invertible.

A pair of dice is thrown. Let E be the event that the sum is greater than or equal to 10 and F be the event "5 appears on the first-die". Find $\text{P}\Big(\frac{\text{E}}{\text{F}}\Big)$. If F is the event "5 appears on at least one die", find $\text{P}\Big(\frac{\text{E}}{\text{F}}\Big)$.

There are three coins. One is two-headed coin (having head on both faces), another is biased coin that comes up heads 75% of the times and third is also a biased coin that comes up tail 40% of the times. One of the three coins is chosen at random and tossed, and it shows heads. What is the probability that it was the two-headed coin?

Evaluvate the following intregals:
$\int\frac{2\tan\text{x}+3}{3\tan\text{x}+4}\ \text{dx}$